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<p><strong>Description:</strong> An introduction to graph theory basics and intuition with applications to scheduling, coloring, and even sexual promiscuity.</p> <p><strong>Speaker:</strong> Tom Leighton</p>
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<p><span m="340">The</span> <span m="450">following</span> <span m="890">content</span> <span m="1480">is</span> <span m="1600">provided</span> <span m="2040">under</span> <span m="2320">a</span> <span m="2360">Creative</span> <span m="2760">Commons</span> <span m="3170">license.</span> <span m="4280">Your</span> <span m="4470">support</span> <span m="4970">will</span> <span m="5130">help</span> <span m="5370">MIT</span> <span m="5830">OpenCourseWare</span> <span m="6620">continue</span> <span m="7130">to</span> <span m="7210">offer</span> <span m="7620">high</span> <span m="7860">quality</span> <span m="8380">educational</span> <span m="9010">resources</span> <span m="9630">for</span> <span m="9780">free.</span> <span m="10980">To</span> <span m="10990">make</span> <span m="11200">a</span> <span m="11240">donation</span> <span m="11930">or</span> <span m="12200">view</span> <span m="12630">additional</span> <span m="13060">materials</span> <span m="13600">from</span> <span m="13750">hundreds</span> <span m="14180">of</span> <span m="14290">MIT</span> <span m="14720">courses,</span> <span m="15820">visit</span> <span m="16040">MIT</span> <span m="16470">OpenCourseWare</span> <span m="17510">at</span> <span m="17680">ocw.mit.edu.</span> </p>
<p><span m="23580">PROFESSOR: OK</span> <span m="24130">let's</span> <span m="24380">get</span> <span m="24550">started.</span> <span m="27790">I</span> <span m="28860">once</span> <span m="29250">taught</span> <span m="29840">with</span> <span m="30110">a</span> <span m="30180">professor</span> <span m="31170">who</span> <span m="31480">was</span> <span m="32070">lamenting</span> <span m="32570">the</span> <span m="32680">fact</span> <span m="33150">that</span> <span m="33260">as</span> <span m="33380">the</span> <span m="33510">term</span> <span m="33820">progresses</span> <span m="35440">attendance</span> <span m="36100">in</span> <span m="36190">lecture</span> <span m="36620">tends</span> <span m="36910">to</span> <span m="37020">drop</span> <span m="37360">off.</span> <span m="37830">And</span> <span m="38670">gets</span> <span m="38890">pretty</span> <span m="39150">dramatic</span> <span m="39620">by</span> <span m="39760">the</span> <span m="39910">end</span> <span m="40050">of</span> <span m="40090">the</span> <span m="40200">term</span> <span m="40640">when</span> <span m="41020">you're</span> <span m="41170">lecturing,</span> <span m="41710">and</span> <span m="42170">nobody's</span> <span m="42620">there.</span> <span m="43760">And</span> <span m="43910">I</span> <span m="43980">asked</span> <span m="44220">him</span> <span m="44370">what</span> <span m="44510">he</span> <span m="44610">did</span> <span m="44820">about</span> <span m="45170">it.</span> <span m="45980">And</span> <span m="46110">he</span> <span m="46230">thought</span> <span m="46490">about</span> <span m="46830">it</span> <span m="47000">and</span> <span m="47310">he</span> <span m="47620">said,</span> <span m="48180">there's</span> <span m="48370">only</span> <span m="48700">two</span> <span m="49010">things</span> <span m="49285">that</span> <span m="49690">can</span> <span m="49930">get</span> <span m="50200">students</span> <span m="50570">to</span> <span m="50690">come</span> <span m="51030">to</span> <span m="51160">lecture,</span> <span m="52480">candy</span> <span m="53660">and</span> <span m="53880">sex.</span> </p>
<p><span m="55840">Now</span> <span m="56830">we've</span> <span m="57090">already</span> <span m="57420">tried</span> <span m="57980">candy,</span> <span m="60380">so</span> <span m="60980">today</span> <span m="61370">we're</span> <span m="61510">going</span> <span m="61610">to</span> <span m="61690">talk</span> <span m="61970">about</span> <span m="62300">sex.</span> <span m="64410">In</span> <span m="64540">fact</span> <span m="65370">we're</span> <span m="65590">going</span> <span m="65700">to</span> <span m="65740">use</span> <span m="66090">graph</span> <span m="66530">theory</span> <span m="67600">to</span> <span m="67740">address</span> <span m="68300">a</span> <span m="68420">decades</span> <span m="69060">old</span> <span m="69360">debate</span> <span m="70050">concerning</span> <span m="70630">the</span> <span m="70770">relative</span> <span m="71380">promiscuity</span> <span m="72290">of</span> <span m="72380">men</span> <span m="72940">versus</span> <span m="73400">women.</span> <span m="75240">Now</span> <span m="76310">graphs</span> <span m="76830">are</span> <span m="76930">incredibly</span> <span m="77610">useful</span> <span m="78020">structures</span> <span m="78470">in</span> <span m="78550">computer</span> <span m="78930">science,</span> <span m="79410">and</span> <span m="79490">we're</span> <span m="79620">going</span> <span m="79720">to</span> <span m="79770">be</span> <span m="79900">studying</span> <span m="80400">them</span> <span m="80690">for</span> <span m="80780">the</span> <span m="80880">next</span> <span m="81660">five</span> <span m="81900">or</span> <span m="82000">six</span> <span m="82270">lectures.</span> <span m="83520">They</span> <span m="83690">come</span> <span m="83860">up</span> <span m="83980">in</span> <span m="84070">all</span> <span m="84300">sorts</span> <span m="84610">of</span> <span m="84710">applications,</span> <span m="86010">scheduling,</span> <span m="87800">optimization,</span> <span m="89040">communications,</span> <span m="90510">the</span> <span m="90620">design</span> <span m="91070">and</span> <span m="91140">analysis</span> <span m="91710">of</span> <span m="91810">algorithms.</span> <span m="92710">In</span> <span m="92820">fact</span> <span m="93090">next</span> <span m="93420">week,</span> <span m="93820">you're</span> <span m="93840">going</span> <span m="93950">to</span> <span m="94020">see</span> <span m="94280">how</span> <span m="94490">to</span> <span m="94750">Stanford</span> <span m="95310">graduate</span> <span m="95770">students</span> <span m="96600">became</span> <span m="97320">gazillionaires</span> <span m="98990">because</span> <span m="99440">they</span> <span m="99570">use</span> <span m="99850">graph</span> <span m="100160">theoretic</span> <span m="100600">in</span> <span m="100920">a</span> <span m="101260">clever</span> <span m="101650">way.</span> </p>
<p><span m="103200">But</span> <span m="103850">let's</span> <span m="104090">talk</span> <span m="104290">about</span> <span m="104560">sex.</span> <span m="106470">The</span> <span m="106640">issue</span> <span m="106910">that</span> <span m="107070">we're</span> <span m="107220">going</span> <span m="107410">to</span> <span m="107780">address</span> <span m="108190">today</span> <span m="108640">is</span> <span m="108770">one</span> <span m="108950">of</span> <span m="109010">the</span> <span m="109090">most</span> <span m="109560">talked</span> <span m="109920">about,</span> <span m="110450">and</span> <span m="110610">most</span> <span m="110940">well</span> <span m="111360">studied,</span> <span m="112660">questions</span> <span m="113270">in</span> <span m="113390">all</span> <span m="113600">of</span> <span m="113690">human</span> <span m="114030">sociology.</span> <span m="115900">On</span> <span m="116200">average,</span> <span m="117540">who</span> <span m="117870">has</span> <span m="118140">more</span> <span m="118550">opposite</span> <span m="119120">gender</span> <span m="119620">partners,</span> <span m="120400">men</span> <span m="121640">or</span> <span m="121810">women?</span> <span m="122715">Now</span> <span m="123210">opposite</span> <span m="123560">gender</span> <span m="124020">is</span> <span m="124520">going</span> <span m="124650">to</span> <span m="124710">be</span> <span m="124840">important.</span> <span m="125310">And</span> <span m="125390">by</span> <span m="125520">this</span> <span m="125770">I</span> <span m="125850">mean,</span> <span m="126540">one</span> <span m="126910">boy,</span> <span m="127740">and</span> <span m="127930">one</span> <span m="128130">girl.</span> <span m="129490">All</span> <span m="129620">right,</span> <span m="129750">I'm</span> <span m="129840">not</span> <span m="130009">making</span> <span m="130289">a</span> <span m="130350">political</span> <span m="130770">statement.</span> <span m="131035">It's</span> <span m="131300">just</span> <span m="131500">that</span> <span m="131760">the</span> <span m="131840">math</span> <span m="132120">is</span> <span m="132230">a</span> <span m="132290">lot</span> <span m="132480">easier</span> <span m="132810">that</span> <span m="133020">way,</span> <span m="133730">as</span> <span m="133970">you'll</span> <span m="134140">see.</span> </p>
<p><span m="136770">Now</span> <span m="136980">I'd</span> <span m="137100">like</span> <span m="137250">to</span> <span m="137360">start</span> <span m="137640">by</span> <span m="137750">taking</span> <span m="138110">a</span> <span m="138180">pole</span> <span m="138580">here</span> <span m="138860">to</span> <span m="138940">see</span> <span m="139200">what</span> <span m="139360">you</span> <span m="139540">think</span> <span m="139790">about</span> <span m="140100">that.</span> <span m="141270">So</span> <span m="141510">raise</span> <span m="141860">your</span> <span m="142260">and</span> <span m="142580">if</span> <span m="142690">you</span> <span m="142890">think</span> <span m="143230">men,</span> <span m="143810">on</span> <span m="144040">average,</span> <span m="144830">have</span> <span m="145050">more</span> <span m="145470">opposite</span> <span m="145980">gender</span> <span m="146330">partners</span> <span m="147160">than</span> <span m="147310">women</span> <span m="147620">do.</span> <span m="150420">Only</span> <span m="150790">a</span> <span m="150840">few.</span> </p>
<p><span m="151095">AUDIENCE: In</span> <span m="151350">life</span> <span m="151780">or</span> <span m="152210">[INAUDIBLE]</span> </p>
<p><span m="153500">PROFESSOR: Um,</span> <span m="153530">you</span> <span m="153965">can--</span> </p>
<p><span m="154400">[LAUGHTER]</span> </p>
<p><span m="159040">PROFESSOR: One</span> <span m="159760">on</span> <span m="160050">one.</span> <span m="160640">OK,</span> <span m="161030">so</span> <span m="161500">let's</span> <span m="161710">say</span> <span m="161880">over</span> <span m="162140">the</span> <span m="162260">course</span> <span m="162600">of</span> <span m="162680">their</span> <span m="162810">lives,</span> <span m="163380">let's</span> <span m="163610">say,</span> <span m="163970">or</span> <span m="164090">over</span> <span m="164210">the</span> <span m="164300">course</span> <span m="164680">of</span> <span m="165150">2010,</span> <span m="166950">that</span> <span m="167100">men</span> <span m="167510">in</span> <span m="167600">America</span> <span m="168200">have</span> <span m="168420">more</span> <span m="168810">opposite-gender</span> <span m="169530">partners</span> <span m="170160">than</span> <span m="170340">women</span> <span m="170700">in</span> <span m="170810">America,</span> <span m="171460">say</span> <span m="171680">in</span> <span m="171790">2010.</span> <span m="172830">Raise</span> <span m="173080">your</span> <span m="173200">hand</span> <span m="173390">if</span> <span m="173450">you</span> <span m="173550">think</span> <span m="173770">men</span> <span m="174010">have</span> <span m="174210">more</span> <span m="174490">going</span> <span m="174760">on.</span> <span m="175030">All</span> <span m="175410">right</span> <span m="175790">a</span> <span m="176170">bunch</span> <span m="176550">of</span> <span m="176780">you.</span> <span m="177040">Raise</span> <span m="177310">your</span> <span m="177460">hand</span> <span m="177700">if</span> <span m="177800">you</span> <span m="177890">think</span> <span m="178070">women</span> <span m="178520">have</span> <span m="178730">more</span> <span m="179450">opposite-gender</span> <span m="180150">partners?</span> <span m="180890">This</span> <span m="181880">is</span> <span m="182000">unusual.</span> <span m="182490">Maybe</span> <span m="182710">even</span> <span m="182900">more</span> <span m="183240">voted</span> <span m="183490">for</span> <span m="183620">women,</span> <span m="183940">but</span> <span m="184060">it's</span> <span m="184190">close.</span> </p>
<p><span m="184870">Raise</span> <span m="185190">your</span> <span m="185340">hand</span> <span m="185680">if</span> <span m="185770">you</span> <span m="185890">think</span> <span m="186110">it's</span> <span m="186240">equal.</span> <span m="187744">All</span> <span m="188191">right,</span> <span m="188640">about</span> <span m="189130">the</span> <span m="189270">same.</span> <span m="189680">Raise</span> <span m="189960">your</span> <span m="190110">hand</span> <span m="190380">if</span> <span m="190450">you</span> <span m="190560">think</span> <span m="190730">there's</span> <span m="190910">no</span> <span m="191140">way</span> <span m="191310">to</span> <span m="191420">know,</span> <span m="192550">that</span> <span m="193220">it's</span> <span m="193560">hopeless</span> <span m="193880">to</span> <span m="193990">really</span> <span m="194220">figure</span> <span m="194470">it</span> <span m="194540">out.</span> <span m="195210">All</span> <span m="195670">right,</span> <span m="196130">nobody</span> <span m="196620">goes</span> <span m="196860">for</span> <span m="196950">that.</span> <span m="197210">All</span> <span m="197515">right,</span> <span m="197820">good.</span> </p>
<p><span m="198565">All</span> <span m="198880">right</span> <span m="199000">well</span> <span m="199160">now</span> <span m="199300">in</span> <span m="199370">the</span> <span m="199470">popular</span> <span m="199940">literature,</span> <span m="200460">I</span> <span m="200510">think</span> <span m="200710">the</span> <span m="200800">feelings</span> <span m="201220">are</span> <span m="201280">different</span> <span m="201730">than</span> <span m="202270">expressed</span> <span m="202800">here.</span> <span m="203620">Pretty</span> <span m="203840">much</span> <span m="204040">universally,</span> <span m="204850">in</span> <span m="204920">the</span> <span m="205000">literature,</span> <span m="205450">it's</span> <span m="205570">believed</span> <span m="205970">that</span> <span m="206070">men</span> <span m="206450">have</span> <span m="206640">more</span> <span m="206890">opposite-gender</span> <span m="207540">partners</span> <span m="208000">than</span> <span m="208130">women.</span> <span m="209320">And</span> <span m="209520">in</span> <span m="209620">fact,</span> <span m="209860">you</span> <span m="209950">could</span> <span m="210070">even</span> <span m="210280">think</span> <span m="210590">about</span> <span m="210900">that,</span> <span m="211140">if</span> <span m="211210">you</span> <span m="211310">think</span> <span m="211500">about</span> <span m="211730">literature,</span> <span m="213120">the</span> <span m="213250">leader</span> <span m="213610">of</span> <span m="213680">the</span> <span m="213760">harem</span> <span m="214270">is</span> <span m="214410">always</span> <span m="214730">a</span> <span m="214790">man.</span> <span m="215780">And</span> <span m="215940">he's</span> <span m="216080">got</span> <span m="216270">lots</span> <span m="216540">of</span> <span m="216630">women.</span> <span m="217680">In</span> <span m="217990">polygamist</span> <span m="218600">cultures,</span> <span m="219780">it's</span> <span m="220140">always</span> <span m="220540">the</span> <span m="220640">man</span> <span m="220990">that</span> <span m="221080">has</span> <span m="221250">multiple</span> <span m="221650">wives,</span> <span m="222380">not</span> <span m="222750">the</span> <span m="222890">reverse.</span> </p>
<p><span m="224890">Now</span> <span m="225700">not</span> <span m="225950">surprisingly,</span> <span m="226700">this</span> <span m="226910">issue</span> <span m="227300">has</span> <span m="227410">been</span> <span m="227560">studied</span> <span m="228830">"scientifically,"</span> <span m="229530">I'll</span> <span m="229620">put</span> <span m="229790">in</span> <span m="229890">quotes,</span> <span m="231730">extensively,</span> <span m="232380">in</span> <span m="232570">one</span> <span m="232730">of</span> <span m="232770">the</span> <span m="232840">largest</span> <span m="233410">studies</span> <span m="234230">ever</span> <span m="234520">done.</span> <span m="236220">Researchers</span> <span m="236850">from</span> <span m="236970">University</span> <span m="237480">of</span> <span m="237550">Chicago</span> <span m="238210">interviewed</span> <span m="239270">2,500</span> <span m="240380">people,</span> <span m="240880">at</span> <span m="241120">random,</span> <span m="241920">over</span> <span m="242190">several</span> <span m="242680">years.</span> <span m="243200">They</span> <span m="243290">brought</span> <span m="243590">them</span> <span m="243730">in,</span> <span m="244560">on</span> <span m="244740">many</span> <span m="245020">occasions,</span> <span m="245800">to</span> <span m="245980">try</span> <span m="246220">to</span> <span m="246280">get</span> <span m="246470">the</span> <span m="246630">answer</span> <span m="247030">for</span> <span m="247150">the</span> <span m="247240">question</span> <span m="247660">once</span> <span m="247900">and</span> <span m="248000">for</span> <span m="248180">all.</span> <span m="248920">And</span> <span m="249140">they</span> <span m="249240">wrote</span> <span m="249520">this</span> <span m="249800">700</span> <span m="250440">page</span> <span m="250790">book,</span> <span m="251560">called</span> <span m="251930">The</span> <span m="252030">soul</span> <span m="252360">of</span> <span m="252430"><i>Social</i></span> <span m="252820"><i>Organization</i></span> <span m="253440"><i>of</i></span> <span m="253520"><i>Sexuality</i>:</span> <span m="254680"><i>Sexual</i></span> <span m="255220"><i>Practices</i></span> <span m="255830"><i>in</i></span> <span m="255900"><i>the</i></span> <span m="255990"><i>US</i>.</span> </p>
<p><span m="258050">Actually</span> <span m="258410">walking</span> <span m="258850">around</span> <span m="259149">with</span> <span m="259240">this</span> <span m="259410">book</span> <span m="259670">has</span> <span m="259800">proved</span> <span m="260060">to</span> <span m="260130">be</span> <span m="260290">a</span> <span m="260350">little</span> <span m="260589">embarrassing.</span> <span m="262720">Last</span> <span m="263140">week</span> <span m="263420">my</span> <span m="263600">11-year-old</span> <span m="264360">daughter</span> <span m="264760">saw</span> <span m="265060">it,</span> <span m="265890">and</span> <span m="266090">she</span> <span m="266220">goes</span> <span m="266520">dad,</span> <span m="266900">why</span> <span m="267350">do</span> <span m="267430">you</span> <span m="267530">have</span> <span m="267800">this</span> <span m="268100">sex</span> <span m="268400">book.</span> <span m="269410">And</span> <span m="270400">I</span> <span m="270520">grabbed</span> <span m="270930">it</span> <span m="271010">back</span> <span m="271400">and</span> <span m="271500">said,</span> <span m="271550">well</span> <span m="271800">that's</span> <span m="272050">for</span> <span m="272120">the</span> <span m="272230">course.</span> <span m="272700">I'm</span> <span m="272740">teaching.</span> <span m="274910">And</span> <span m="275550">I</span> <span m="275640">thought</span> <span m="275860">I'd</span> <span m="275990">gotten</span> <span m="276260">away</span> <span m="276550">with</span> <span m="276740">it,</span> <span m="277040">and</span> <span m="277130">everything</span> <span m="277890">was</span> <span m="278050">fine.</span> <span m="278930">And</span> <span m="279180">then</span> <span m="279610">later</span> <span m="280000">that</span> <span m="280250">day</span> <span m="280530">she</span> <span m="280730">texted</span> <span m="281150">all</span> <span m="281420">of</span> <span m="281530">our</span> <span m="281620">friends</span> <span m="282150">about</span> <span m="282490">the</span> <span m="282580">new</span> <span m="282810">news</span> <span m="283290">that</span> <span m="283660">what</span> <span m="283930">do</span> <span m="283980">you</span> <span m="284070">know,</span> <span m="284330">her</span> <span m="284480">dad</span> <span m="284880">teaches</span> <span m="285210">sex</span> <span m="285520">ed</span> <span m="285860">at</span> <span m="286180">MIT.</span> </p>
<p><span m="288960">Anyway</span> <span m="289440">this</span> <span m="289690">study</span> <span m="290090">concludes</span> <span m="291880">that</span> <span m="292820">on</span> <span m="293070">average</span> <span m="293540">men</span> <span m="293950">have</span> <span m="294190">74%</span> <span m="295610">more</span> <span m="296400">opposite-gender</span> <span m="297230">partners</span> <span m="297710">than</span> <span m="297850">women.</span> <span m="300430">There's</span> <span m="300540">one</span> <span m="300700">other</span> <span m="300910">central</span> <span m="301610">claims.</span> </p>
<p></p>
<p><span m="310420">And</span> <span m="310530">this</span> <span m="310670">is</span> <span m="310780">in</span> <span m="310870">the</span> <span m="310980">US.</span> </p>
<p></p>
<p><span m="336420">OK</span> <span m="337100">now,</span> <span m="338250">when</span> <span m="338300">you</span> <span m="338400">think</span> <span m="338680">about</span> <span m="339010">it</span> <span m="339616">that</span> <span m="340310">sounds</span> <span m="341050">maybe</span> <span m="341750">reasonable,</span> <span m="343350">might</span> <span m="343880">be</span> <span m="344010">OK.</span> <span m="345390">But</span> <span m="346330">not</span> <span m="346590">according</span> <span m="346990">to</span> <span m="347100">ABC</span> <span m="347610">News.</span> <span m="349750">They</span> <span m="350010">did</span> <span m="350180">a</span> <span m="350240">poll</span> <span m="350670">of</span> <span m="350770">1,500</span> <span m="351780">people</span> <span m="352110">in</span> <span m="352160">the</span> <span m="352250">country,</span> <span m="352720">in</span> <span m="352860">2004,</span> <span m="354315">and</span> <span m="354750">concluded</span> <span m="355320">that</span> <span m="355470">the</span> <span m="356180">average</span> <span m="356660">disparity</span> <span m="357330">is</span> <span m="357470">much</span> <span m="357990">greater.</span> <span m="359090">In</span> <span m="359210">particular,</span> <span m="360560">in</span> <span m="360600">this</span> <span m="360790">study,</span> <span m="361740">they</span> <span m="361950">said</span> <span m="362170">that</span> <span m="362230">the</span> <span m="362440">average</span> <span m="362860">man</span> <span m="363280">has</span> <span m="363590">20</span> <span m="364000">partners--</span> <span m="365210">I'm</span> <span m="365420">assuming</span> <span m="365770">over</span> <span m="365950">their</span> <span m="366080">lifetime--</span> <span m="367100">and</span> <span m="367590">the</span> <span m="367730">average</span> <span m="368090">woman</span> <span m="368440">has</span> <span m="368740">six.</span> <span m="370050">And</span> <span m="370300">this</span> <span m="370480">gives</span> <span m="370680">a</span> <span m="370740">disparity</span> <span m="372160">233%.</span> </p>
<p><span m="381220">So</span> <span m="381390">ABC</span> <span m="381940">News</span> <span m="382740">did</span> <span m="383470">a</span> <span m="383530">smaller</span> <span m="384020">survey</span> <span m="385540">says</span> <span m="386050">that</span> <span m="386160">it's</span> <span m="386400">233%</span> <span m="388100">here,</span> <span m="390630">much</span> <span m="390950">more</span> <span m="391220">than</span> <span m="391570">74%.</span> <span m="393540">Now</span> <span m="393790">ABC</span> <span m="394290">News</span> <span m="394720">claimed</span> <span m="396000">this</span> <span m="396210">is</span> <span m="396330">one</span> <span m="396490">of</span> <span m="396530">the</span> <span m="396580">most</span> <span m="396800">scientific</span> <span m="397580">studies</span> <span m="397970">ever</span> <span m="398180">done.</span> <span m="399060">And</span> <span m="399310">there</span> <span m="399430">was</span> <span m="399600">a</span> <span m="399730">2.5%</span> <span m="400820">margin</span> <span m="401210">of</span> <span m="401320">error.</span> <span m="402430">Now</span> <span m="402590">we'll</span> <span m="402760">actually</span> <span m="403390">talk</span> <span m="403750">about</span> <span m="404180">what</span> <span m="404320">that</span> <span m="404520">means</span> <span m="404840">mathematically</span> <span m="405540">later</span> <span m="405810">in</span> <span m="405860">the</span> <span m="405960">term</span> <span m="406310">when</span> <span m="406390">we</span> <span m="406520">do</span> <span m="406860">probability,</span> <span m="407200">and</span> <span m="407320">do</span> <span m="407570">study</span> <span m="407850">polling.</span> <span m="409550">Now</span> <span m="409960">of</span> <span m="410000">course</span> <span m="410220">I</span> <span m="410250">should</span> <span m="410430">also</span> <span m="410680">mention</span> <span m="410970">that</span> <span m="411080">ABC</span> <span m="411510">News</span> <span m="411770">is</span> <span m="411890">the</span> <span m="411990">one</span> <span m="412260">that</span> <span m="412410">said</span> <span m="412770">Al</span> <span m="413020">Gore</span> <span m="413390">won</span> <span m="413650">the</span> <span m="413730">presidential</span> <span m="414350">election</span> <span m="414710">in</span> <span m="416180">2000.</span> </p>
<p><span m="417720">Now</span> <span m="418790">the</span> <span m="419010">study</span> <span m="419450">is</span> <span m="419590">called</span> <span m="419920">American</span> <span m="420420">Sex</span> <span m="420840">Survey,</span> <span m="421930">a</span> <span m="422040">Peak</span> <span m="422410">Between</span> <span m="422850">the</span> <span m="422920">Sheets.</span> <span m="424740">That</span> <span m="424790">doesn't</span> <span m="425070">sound</span> <span m="425410">so</span> <span m="425530">scientific.</span> <span m="426530">And</span> <span m="426710">it</span> <span m="426750">was</span> <span m="426900">on</span> <span m="427090">TV,</span> <span m="427520">on</span> <span m="427640">Primetime</span> <span m="428190">Live</span> <span m="428520">in</span> <span m="428590">2004.</span> <span m="429990">The</span> <span m="430110">promo</span> <span m="430630">for</span> <span m="430770">this</span> <span m="430990">is</span> <span m="431100">really</span> <span m="431360">good.</span> <span m="432300">It</span> <span m="432460">says,</span> <span m="432810">a</span> <span m="433110">groundbreaking</span> <span m="434090">ABC</span> <span m="434590">News</span> <span m="434880">Primetime</span> <span m="435500">Live</span> <span m="435800">survey</span> <span m="436740">finds</span> <span m="437060">a</span> <span m="437160">range</span> <span m="437500">of</span> <span m="437600">eye</span> <span m="437830">popping</span> <span m="438270">sexual</span> <span m="438740">activities,</span> <span m="439330">fantasies,</span> <span m="440030">and</span> <span m="440180">attitudes</span> <span m="440740">in</span> <span m="440800">this</span> <span m="440980">country,</span> <span m="441910">confirming</span> <span m="442460">some</span> <span m="442720">conventional</span> <span m="443340">wisdom,</span> <span m="444070">exploding</span> <span m="444440">some</span> <span m="444810">myths,</span> <span m="445700">and</span> <span m="445960">venturing</span> <span m="446240">where</span> <span m="446520">few</span> <span m="447110">scientific</span> <span m="447940">surveys</span> <span m="448430">have</span> <span m="448560">gone</span> <span m="448860">before.</span> <span m="450150">By</span> <span m="450310">the</span> <span m="450410">end</span> <span m="450540">of</span> <span m="450620">today,</span> <span m="450930">we're</span> <span m="451120">going</span> <span m="451220">to</span> <span m="451390">agree</span> <span m="451680">with</span> <span m="451780">that</span> <span m="451940">last</span> <span m="452200">statement.</span> </p>
<p><span m="454390">OK</span> <span m="456280">now</span> <span m="456520">who</span> <span m="456630">do</span> <span m="456700">you</span> <span m="456780">think's</span> <span m="457050">right?</span> <span m="459250">University</span> <span m="459800">of</span> <span m="459870">Chicago.</span> <span m="461980">Who</span> <span m="462140">votes</span> <span m="462430">for</span> <span m="462540">74%</span> <span m="463480">as</span> <span m="463590">being</span> <span m="463800">pretty</span> <span m="464040">close?</span> <span m="465010">A</span> <span m="465410">few</span> <span m="465810">of</span> <span m="465910">you.</span> <span m="467190">I've</span> <span m="467420">already</span> <span m="467940">slammed</span> <span m="468360">these</span> <span m="468610">guys.</span> <span m="468890">Who</span> <span m="469000">votes</span> <span m="469280">for</span> <span m="469410">ABC</span> <span m="469850">News</span> <span m="470310">as</span> <span m="470440">being</span> <span m="470670">more</span> <span m="470930">accurate?</span> <span m="471830">Yeah,</span> <span m="472710">nobody.</span> <span m="474980">Who</span> <span m="475130">votes</span> <span m="475430">for</span> <span m="475550">no</span> <span m="475830">way</span> <span m="476000">to</span> <span m="476120">tell?</span> <span m="478210">I</span> <span m="478260">got</span> <span m="478460">some</span> <span m="478700">votes</span> <span m="479030">there,</span> <span m="479370">all</span> <span m="479420">right.</span> <span m="480360">So</span> <span m="480490">how</span> <span m="480720">do</span> <span m="480820">you</span> <span m="480920">tackle</span> <span m="481320">this</span> <span m="481500">problem?</span> </p>
<p><span m="483230">In</span> <span m="483330">theory</span> <span m="483650">we</span> <span m="483760">could</span> <span m="483890">do</span> <span m="484010">our</span> <span m="484200">own</span> <span m="484340">6.042</span> <span m="485130">survey.</span> <span m="486510">I</span> <span m="486650">don't</span> <span m="486830">know</span> <span m="487010">how</span> <span m="487160">much</span> <span m="487360">we'd</span> <span m="487500">really</span> <span m="487780">learn,</span> <span m="488080">and</span> <span m="488200">for</span> <span m="488500">sure</span> <span m="488600">I'd</span> <span m="488830">get</span> <span m="489110">fired.</span> <span m="490430">So</span> <span m="491180">I</span> <span m="491300">don't</span> <span m="491410">think</span> <span m="491540">we're</span> <span m="491650">going</span> <span m="491760">to</span> <span m="491850">do</span> <span m="492010">that.</span> <span m="494030">But</span> <span m="494230">fortunately,</span> <span m="495090">this</span> <span m="495320">is</span> <span m="495420">the</span> <span m="495520">kind</span> <span m="495750">of</span> <span m="495830">question</span> <span m="496310">that</span> <span m="496400">could</span> <span m="496520">be</span> <span m="496640">handled,</span> <span m="497360">and</span> <span m="497560">actually</span> <span m="497940">answered,</span> <span m="498880">by</span> <span m="499000">graph</span> <span m="499330">theory,</span> <span m="500960">even</span> <span m="501070">though</span> <span m="501200">it</span> <span m="501260">might</span> <span m="501430">be</span> <span m="501530">more</span> <span m="501750">interesting</span> <span m="502230">to</span> <span m="502560">interview</span> <span m="502730">thousands</span> <span m="503190">of</span> <span m="503300">people,</span> <span m="503680">and</span> <span m="503910">find</span> <span m="504210">out</span> <span m="504300">what's</span> <span m="504470">going</span> <span m="504690">on.</span> <span m="505640">That's</span> <span m="505900">not</span> <span m="506130">as</span> <span m="506230">efficient</span> <span m="506700">as</span> <span m="507370">using</span> <span m="507640">graphs.</span> </p>
<p><span m="508760">So</span> <span m="508800">let</span> <span m="508940">me</span> <span m="509050">start</span> <span m="509590">by</span> <span m="509730">defining</span> <span m="510360">what</span> <span m="510560">a</span> <span m="510610">graph</span> <span m="510980">is.</span> <span m="513070">Informally</span> <span m="513770">graph</span> <span m="514380">is</span> <span m="514650">just</span> <span m="514909">a</span> <span m="514970">bunch</span> <span m="515260">of</span> <span m="515429">dots</span> <span m="516169">and</span> <span m="516360">lines</span> <span m="517210">connecting</span> <span m="517610">the</span> <span m="517700">dots,</span> <span m="518110">it's</span> <span m="518299">actually</span> <span m="518600">very</span> <span m="518830">simple.</span> <span m="520409">So</span> <span m="520570">here's</span> <span m="520809">to</span> <span m="520870">graph.</span> <span m="526830">These</span> <span m="527170">are</span> <span m="527300">the</span> <span m="527430">nodes,</span> <span m="528450">and</span> <span m="528650">they're</span> <span m="528780">connected</span> <span m="529460">with</span> <span m="529570">these</span> <span m="529750">lines,</span> <span m="530530">called</span> <span m="530930">edges.</span> <span m="535480">And</span> <span m="535670">often</span> <span m="535970">the</span> <span m="536070">nodes,</span> <span m="536590">and</span> <span m="536730">sometimes</span> <span m="537170">the</span> <span m="537270">edges,</span> <span m="537530">are</span> <span m="537600">labeled.</span> <span m="538770">For</span> <span m="538910">example,</span> <span m="539450">we</span> <span m="539460">might</span> <span m="539650">call</span> <span m="539870">this</span> <span m="540080">x1,</span> <span m="543930">x2,</span> <span m="545900">x3,</span> <span m="547720">x4,</span> <span m="549180">x5,</span> <span m="549510">x6,</span> <span m="549740">and</span> <span m="549890">x7.</span> <span m="550323">So</span> <span m="551190">that's</span> <span m="551460">an</span> <span m="551510">example</span> <span m="551990">of</span> <span m="552050">a</span> <span m="552090">graph.</span> </p>
<p><span m="553090">Now</span> <span m="553650">this</span> <span m="553800">being</span> <span m="554010">a</span> <span m="554050">math</span> <span m="554360">class,</span> <span m="554740">we</span> <span m="554820">got</span> <span m="554940">to</span> <span m="554980">give</span> <span m="555120">a</span> <span m="555190">formal</span> <span m="555640">definition</span> <span m="556160">of</span> <span m="556230">a</span> <span m="556270">graph.</span> <span m="557770">And</span> <span m="557950">we'll</span> <span m="558060">usually</span> <span m="558590">use</span> <span m="558790">the</span> <span m="558870">formal</span> <span m="559200">definition.</span> <span m="563910">A</span> <span m="564100">graph</span> <span m="567070">G</span> <span m="569470">is</span> <span m="569860">a</span> <span m="569980">pair</span> <span m="570370">of</span> <span m="570450">sets</span> <span m="575630">often</span> <span m="576060">called</span> <span m="576440">V</span> <span m="576850">and</span> <span m="577345">E.</span> <span m="581310">Where</span> <span m="581510">V</span> <span m="582200">is</span> <span m="582350">a</span> <span m="582410">set</span> <span m="582660">of</span> <span m="583350">elements</span> <span m="583730">called</span> <span m="583970">vertices</span> <span m="584460">or</span> <span m="584540">nodes.</span> <span m="585850">And</span> <span m="586050">it</span> <span m="586100">has</span> <span m="586380">to</span> <span m="586450">be</span> <span m="586550">non-empty</span> <span m="587100">here</span> <span m="587510">in</span> <span m="587710">this</span> <span m="587860">class.</span> </p>
<p></p>
<p><span m="606140">And</span> <span m="606410">we'll</span> <span m="606560">go</span> <span m="606730">back</span> <span m="607030">and</span> <span m="607150">forth</span> <span m="607450">between</span> <span m="607760">vertices</span> <span m="608310">and</span> <span m="608360">nodes.</span> <span m="608820">Even</span> <span m="609060">the</span> <span m="609160">text</span> <span m="610080">we</span> <span m="610240">use</span> <span m="610450">both</span> <span m="610720">words</span> <span m="611020">interchangeably.</span> <span m="613270">And</span> <span m="613550">E</span> <span m="615850">is</span> <span m="616120">a</span> <span m="616650">set</span> <span m="618320">of</span> <span m="619230">2-item</span> <span m="621670">subsets</span> <span m="625570">and</span> <span m="625910">V,</span> <span m="626590">and</span> <span m="626730">they're</span> <span m="626880">called</span> <span m="627470">edges.</span> </p>
<p><span m="632970">So</span> <span m="633070">for</span> <span m="633210">example,</span> <span m="633770">over</span> <span m="633950">here</span> <span m="634200">in</span> <span m="634260">this</span> <span m="634430">picture,</span> <span m="637060">V</span> <span m="638640">is</span> <span m="638950">the</span> <span m="639030">set</span> <span m="639240">of</span> <span m="639310">nodes</span> <span m="639730">is</span> <span m="640400">x1,</span> <span m="642940">x2,</span> <span m="643880">x3,</span> <span m="645240">up</span> <span m="645655">to</span> <span m="646070">x7,</span> <span m="646720">that's</span> <span m="646940">the</span> <span m="647040">nodes.</span> <span m="648950">And</span> <span m="649200">E,</span> <span m="650400">the</span> <span m="650470">set</span> <span m="650690">of</span> <span m="650790">edges,</span> <span m="652280">is</span> <span m="653030">pairs,</span> <span m="654430">unordered</span> <span m="655070">pairs</span> <span m="656270">of</span> <span m="656440">vertices.</span> <span m="658700">So</span> <span m="658870">for</span> <span m="658990">example</span> <span m="659550">x1,</span> <span m="661310">x2</span> <span m="662490">is</span> <span m="662690">an</span> <span m="662780">edge.</span> <span m="664615">And</span> <span m="665070">it's</span> <span m="665540">the</span> <span m="665600">same</span> <span m="666010">as</span> <span m="666330">the</span> <span m="666430">set</span> <span m="666760">x2,</span> <span m="667210">x1,</span> <span m="667660">doesn't</span> <span m="667910">matter</span> <span m="668180">the</span> <span m="668330">order</span> <span m="668590">here.</span> <span m="669210">Later</span> <span m="669680">in</span> <span m="669720">a</span> <span m="669920">week</span> <span m="670180">or</span> <span m="670260">so,</span> <span m="670400">we'll</span> <span m="670500">talk</span> <span m="670730">about</span> <span m="670910">directed</span> <span m="671240">graphs</span> <span m="671680">where</span> <span m="671790">the</span> <span m="672050">order</span> <span m="672335">matters.</span> <span m="674600">x1,</span> <span m="675980">x3</span> <span m="676430">is</span> <span m="676590">also</span> <span m="676960">an</span> <span m="677040">edge</span> <span m="678790">here,</span> <span m="680750">and</span> <span m="681040">so</span> <span m="681300">one.</span> <span m="681750">Think</span> <span m="682200">we've</span> <span m="682650">got,</span> <span m="682870">let's</span> <span m="683070">see,</span> <span m="683230">1,</span> <span m="683580">2,</span> <span m="683640">3,</span> <span m="683800">4,</span> <span m="684277">5,</span> <span m="684754">6,</span> <span m="685231">7</span> <span m="685710">edges</span> <span m="686010">in</span> <span m="686060">this</span> <span m="686220">graph.</span> <span m="687570">And</span> <span m="688020">the</span> <span m="688120">last</span> <span m="688440">one</span> <span m="688550">would</span> <span m="688690">be</span> <span m="689900">x5,</span> <span m="690980">x7.</span> </p>
<p><span m="695300">Edges</span> <span m="695680">are</span> <span m="695810">also</span> <span m="696320">sometimes</span> <span m="696720">written</span> <span m="696950">with</span> <span m="697070">this</span> <span m="697270">notation,</span> <span m="698760">x1</span> <span m="699350">line</span> <span m="699930">x2,</span> <span m="701190">is</span> <span m="701320">another</span> <span m="701570">notation.</span> <span m="703120">And</span> <span m="703240">then</span> <span m="703420">later</span> <span m="703680">when</span> <span m="703790">you</span> <span m="703860">talk</span> <span m="704060">about</span> <span m="704250">directed</span> <span m="704690">edges,</span> <span m="704960">we'll</span> <span m="705050">put</span> <span m="705180">a</span> <span m="705230">little</span> <span m="705520">arrowhead</span> <span m="706170">on</span> <span m="706520">one</span> <span m="706710">end</span> <span m="706890">of</span> <span m="706970">this.</span> </p>
<p><span m="709710">Now</span> <span m="709980">the</span> <span m="710090">definition</span> <span m="710570">of</span> <span m="710630">a</span> <span m="710680">graph</span> <span m="710990">is</span> <span m="711130">really</span> <span m="711400">pretty</span> <span m="711640">simple.</span> <span m="712110">Just</span> <span m="712350">think</span> <span m="712550">of</span> <span m="712640">it</span> <span m="712680">as</span> <span m="712830">dots</span> <span m="713140">and</span> <span m="713230">lines,</span> <span m="713670">if</span> <span m="713780">you</span> <span m="713890">want.</span> <span m="715290">But</span> <span m="715480">there's</span> <span m="715690">often</span> <span m="716510">differences</span> <span m="717330">in</span> <span m="717410">how</span> <span m="717570">people</span> <span m="717840">define</span> <span m="718220">graphs.</span> <span m="719440">For</span> <span m="719570">example,</span> <span m="721350">in</span> <span m="721520">this</span> <span m="721710">class</span> <span m="722680">we</span> <span m="722910">don't</span> <span m="723130">allow</span> <span m="723480">the</span> <span m="723670">empty</span> <span m="723980">graph,</span> <span m="724550">i.e.</span> <span m="725050">the</span> <span m="725160">graph</span> <span m="725520">with</span> <span m="725640">no</span> <span m="725950">nodes.</span> <span m="726820">So</span> <span m="726920">we're</span> <span m="727060">going</span> <span m="727150">to</span> <span m="727190">insist</span> <span m="727510">that</span> <span m="727830">every</span> <span m="728020">graph</span> <span m="728390">has</span> <span m="728630">to</span> <span m="728720">have</span> <span m="728950">at</span> <span m="729040">least</span> <span m="729310">one</span> <span m="729530">node</span> <span m="729750">in</span> <span m="729850">it.</span> <span m="730550">And</span> <span m="730670">that's</span> <span m="730850">just</span> <span m="731010">to</span> <span m="731080">make</span> <span m="731320">the</span> <span m="731410">theorems</span> <span m="731770">we're</span> <span m="731840">going</span> <span m="731920">prove</span> <span m="731980">be</span> <span m="732370">true.</span> <span m="733010">Otherwise</span> <span m="733350">there's</span> <span m="733440">some</span> <span m="733690">theorems</span> <span m="734150">that</span> <span m="734260">are</span> <span m="734670">false</span> <span m="734740">for</span> <span m="734830">the</span> <span m="735170">special</span> <span m="735450">case</span> <span m="735490">of</span> <span m="735885">the</span> <span m="736280">empty</span> <span m="736580">graph.</span> </p>
<p><span m="736880">But</span> <span m="737020">we</span> <span m="737140">don't</span> <span m="737500">require</span> <span m="737960">the</span> <span m="738080">graph</span> <span m="738430">to</span> <span m="738530">have</span> <span m="738730">any</span> <span m="738920">edges.</span> <span m="739270">In</span> <span m="739620">fact,</span> <span m="740060">it's</span> <span m="740440">possible</span> <span m="740980">you</span> <span m="741060">have</span> <span m="742380">a</span> <span m="742450">graph</span> <span m="742810">with</span> <span m="742960">nodes,</span> <span m="743330">but</span> <span m="743460">no</span> <span m="743660">edges.</span> <span m="746500">For</span> <span m="746630">example,</span> <span m="747050">this</span> <span m="747240">graph.</span> <span m="748560">Three-node</span> <span m="748960">graph.</span> <span m="753560">So</span> <span m="753670">here</span> <span m="753930">G</span> <span m="754420">equals</span> <span m="756240">VE,</span> <span m="757260">V</span> <span m="757760">equals</span> <span m="758810">x1,</span> <span m="759810">x2,</span> <span m="760600">x3.</span> <span m="762730">And</span> <span m="762930">E</span> <span m="763130">is</span> <span m="763220">just</span> <span m="763450">the</span> <span m="763530">empty</span> <span m="763770">set.</span> <span m="768110">Now</span> <span m="768240">for</span> <span m="768410">a</span> <span m="768460">general</span> <span m="768840">graph,</span> <span m="770310">when</span> <span m="770440">you</span> <span m="770530">do</span> <span m="770730">have</span> <span m="770960">edges,</span> <span m="773060">we</span> <span m="773840">say</span> <span m="774150">that</span> <span m="776020">two</span> <span m="776230">nodes,</span> <span m="779560">call</span> <span m="779820">them</span> <span m="780290">xi</span> <span m="780550">and</span> <span m="781080">xj,</span> <span m="784005">are</span> <span m="784440">adjacent</span> <span m="791050">if</span> <span m="791450">they're</span> <span m="791580">connected</span> <span m="792020">by</span> <span m="792220">an</span> <span m="792290">edge,</span> <span m="792670">namely</span> <span m="793120">if</span> <span m="793563">xi</span> <span m="794006">xj</span> <span m="795610">is</span> <span m="795920">an</span> <span m="796010">edge.</span> </p>
<p><span m="798160">All</span> <span m="798270">right</span> <span m="798470">so</span> <span m="799400">for</span> <span m="799510">example,</span> <span m="801130">x5</span> <span m="801990">is</span> <span m="802140">adjacent</span> <span m="802530">to</span> <span m="802620">x7,</span> <span m="804630">but</span> <span m="804850">it's</span> <span m="805050">not</span> <span m="805370">adjacent</span> <span m="806450">to</span> <span m="806530">x4,</span> <span m="808260">there's</span> <span m="808420">no</span> <span m="808620">edge</span> <span m="808810">there.</span> <span m="813160">Closely</span> <span m="813720">related</span> <span m="814170">is</span> <span m="814540">the</span> <span m="814640">definition</span> <span m="815190">of</span> <span m="815250">the</span> <span m="815330">incidence.</span> <span m="817810">An</span> <span m="817920">edge</span> <span m="821280">E,</span> <span m="823390">which</span> <span m="823640">is</span> <span m="824080">xixj,</span> <span m="827950">is</span> <span m="828170">said</span> <span m="828410">to</span> <span m="828480">be</span> <span m="828710">incident</span> <span m="833980">to</span> <span m="834210">its</span> <span m="834340">end</span> <span m="834530">points,</span> <span m="834900">xi</span> <span m="835080">and</span> <span m="835400">xj.</span> <span m="839120">OK</span> <span m="839550">so,</span> <span m="841060">for</span> <span m="841190">example,</span> <span m="843240">if</span> <span m="843400">I</span> <span m="843480">labeled</span> <span m="843940">that</span> <span m="844570">edge</span> <span m="844970">as</span> <span m="845446">E,</span> <span m="845922">E</span> <span m="846400">is</span> <span m="846530">the</span> <span m="846690">edge</span> <span m="847010">x1,</span> <span m="847640">x2,</span> <span m="848590">and</span> <span m="848750">this</span> <span m="848970">incident</span> <span m="849410">to</span> <span m="849490">x1,</span> <span m="850790">and</span> <span m="850940">incident</span> <span m="851620">x2.</span> </p>
<p><span m="858320">Then</span> <span m="858540">we</span> <span m="858660">can</span> <span m="858800">talk</span> <span m="859100">about</span> <span m="859640">the</span> <span m="859800">degree</span> <span m="860160">of</span> <span m="860230">a</span> <span m="860290">node.</span> <span m="863410">The</span> <span m="863570">number</span> <span m="868000">of</span> <span m="868160">edges</span> <span m="870970">incident</span> <span m="875020">to</span> <span m="875270">a</span> <span m="875320">node</span> <span m="878180">is</span> <span m="878600">called</span> <span m="879080">the</span> <span m="879160">degree</span> <span m="879510">of</span> <span m="879580">the</span> <span m="879670">node.</span> </p>
<p></p>
<p><span m="891510">So</span> <span m="891710">for</span> <span m="891840">example,</span> <span m="892380">what's</span> <span m="892540">the</span> <span m="892650">degree</span> <span m="893090">of</span> <span m="893170">x5</span> <span m="894820">over</span> <span m="895090">here?</span> <span m="897520">3,</span> <span m="898050">so</span> <span m="898190">in</span> <span m="898260">this</span> <span m="898450">case,</span> <span m="898920">the</span> <span m="899020">degree</span> <span m="901000">of</span> <span m="901210">x5</span> <span m="902870">equals</span> <span m="903550">3.</span> <span m="905010">The</span> <span m="905120">degree</span> <span m="905480">of</span> <span m="905570">x7</span> <span m="907880">is</span> <span m="908060">1.</span> <span m="910920">These</span> <span m="911190">guys</span> <span m="912210">all</span> <span m="912480">have</span> <span m="912710">degree</span> <span m="913100">0,</span> <span m="914000">there's</span> <span m="914190">no</span> <span m="914410">edges</span> <span m="914900">incident</span> <span m="915350">to</span> <span m="915788">them.</span> </p>
<p><span m="917980">Now</span> <span m="918720">in</span> <span m="918860">this</span> <span m="919060">class,</span> <span m="919420">we're</span> <span m="919570">going</span> <span m="919660">to</span> <span m="919720">look</span> <span m="919920">at</span> <span m="920180">only</span> <span m="920630">simple</span> <span m="921120">graphs,</span> <span m="921790">at</span> <span m="922120">lease</span> <span m="922390">for</span> <span m="922530">a</span> <span m="922600">while.</span> <span m="926570">A</span> <span m="926690">graph</span> <span m="928580">is</span> <span m="928830">simple</span> <span m="933920">if</span> <span m="934100">it</span> <span m="934380">has</span> <span m="935890">no</span> <span m="936600">loops,</span> <span m="939280">or</span> <span m="939920">multiple</span> <span m="940460">edges.</span> <span m="948200">Now</span> <span m="948370">a</span> <span m="948450">loop</span> <span m="949870">is</span> <span m="950470">an</span> <span m="950640">edge</span> <span m="951130">that</span> <span m="951350">only</span> <span m="951660">connects</span> <span m="952040">up</span> <span m="952170">one</span> <span m="952440">node,</span> <span m="953500">that's</span> <span m="953750">a</span> <span m="953800">loop</span> <span m="954140">and</span> <span m="954240">we</span> <span m="954340">don't</span> <span m="954550">allow</span> <span m="954870">it.</span> <span m="956250">A</span> <span m="956350">multiple</span> <span m="956950">edge</span> <span m="957390">is</span> <span m="957550">we've</span> <span m="957800">got</span> <span m="958030">two</span> <span m="958430">edges</span> <span m="958810">that</span> <span m="958920">are</span> <span m="959000">really</span> <span m="959280">the</span> <span m="959390">same,</span> <span m="960070">they</span> <span m="960170">connect</span> <span m="960570">the</span> <span m="960670">same</span> <span m="961010">endpoints.</span> <span m="962980">Also</span> <span m="963280">called</span> <span m="963560">a</span> <span m="963590">multi-edge.</span> <span m="968290">And</span> <span m="968430">those</span> <span m="969120">we're</span> <span m="969220">not</span> <span m="969400">going</span> <span m="969490">to</span> <span m="969540">have</span> <span m="969850">in</span> <span m="969960">simple</span> <span m="970250">graphs.</span> <span m="970620">We</span> <span m="970720">don't</span> <span m="970890">allow</span> <span m="971160">this.</span> <span m="971975">We</span> <span m="972340">don't</span> <span m="972540">allow</span> <span m="972770">that.</span> <span m="975200">Any</span> <span m="975370">questions</span> <span m="976380">so</span> <span m="976560">far</span> <span m="976850">about</span> <span m="977010">what</span> <span m="977120">a</span> <span m="977160">graph</span> <span m="977480">is?</span> </p>
<p><span m="982490">So</span> <span m="983830">how</span> <span m="984160">are</span> <span m="984280">we</span> <span m="984390">going</span> <span m="984510">to</span> <span m="984550">use</span> <span m="984880">a</span> <span m="984990">graph</span> <span m="986390">to</span> <span m="986500">model</span> <span m="987030">the</span> <span m="987130">problem</span> <span m="988820">of</span> <span m="988960">opposite-gender</span> <span m="990400">partners?</span> <span m="992470">That's</span> <span m="992710">the</span> <span m="992790">question</span> <span m="993150">we're</span> <span m="993290">after.</span> <span m="994890">So</span> <span m="996510">any</span> <span m="996760">thoughts</span> <span m="997120">about</span> <span m="997360">what</span> <span m="997510">the</span> <span m="997600">nodes</span> <span m="998000">of</span> <span m="998070">the</span> <span m="998170">graph</span> <span m="998510">are</span> <span m="998570">going</span> <span m="998670">to</span> <span m="998730">represent?</span> <span m="1001840">What</span> <span m="1002010">is</span> <span m="1002120">it?</span> </p>
<p><span m="1002440">AUDIENCE: Males</span> <span m="1002760">and</span> <span m="1003150">females?</span> </p>
<p><span m="1003930">PROFESSOR: People.</span> <span m="1005370">Yeah,</span> <span m="1005760">so</span> <span m="1005900">we're</span> <span m="1006010">going</span> <span m="1006110">to</span> <span m="1006180">have</span> <span m="1007260">people.</span> <span m="1007670">In</span> <span m="1007770">fact,</span> <span m="1007950">there's</span> <span m="1008090">two</span> <span m="1008250">kinds</span> <span m="1008570">of</span> <span m="1008660">people</span> <span m="1008980">here.</span> <span m="1010290">There's</span> <span m="1010460">men,</span> <span m="1011980">and</span> <span m="1012190">women.</span> <span m="1015370">All</span> <span m="1015480">right</span> <span m="1015790">we</span> <span m="1015920">got</span> <span m="1016170">nodes</span> <span m="1016720">here</span> <span m="1018620">for</span> <span m="1018730">the</span> <span m="1018870">men.</span> <span m="1021520">And</span> <span m="1021710">in</span> <span m="1021770">fact</span> <span m="1022020">in</span> <span m="1022060">America,</span> <span m="1022450">there's</span> <span m="1022610">a</span> <span m="1022670">lot</span> <span m="1022890">of</span> <span m="1022960">nodes</span> <span m="1023270">here.</span> <span m="1025400">All</span> <span m="1025660">right,</span> <span m="1025910">and</span> <span m="1026020">so</span> <span m="1026119">this</span> <span m="1026349">might</span> <span m="1026609">be</span> <span m="1027060">oh</span> <span m="1027119">I</span> <span m="1027400">don't</span> <span m="1027560">know,</span> <span m="1027750">say</span> <span m="1028150">that's</span> <span m="1028510">Tom</span> <span m="1028839">Cruise</span> <span m="1029480">and</span> <span m="1030880">Nicole</span> <span m="1032619">Kidman.</span> <span m="1034310">Now</span> <span m="1034460">what's</span> <span m="1034720">the</span> <span m="1035060">edge</span> <span m="1035170">going</span> <span m="1035240">to</span> <span m="1035600">represent?</span> </p>
<p><span m="1036944">AUDIENCE: Partners.</span> </p>
<p><span m="1038420">PROFESSOR: Partners.</span> <span m="1040020">They</span> <span m="1040130">were</span> <span m="1040230">opposite-gender</span> <span m="1041010">partners.</span> <span m="1043109">And</span> <span m="1043400">there's</span> <span m="1043660">actually</span> <span m="1043970">more</span> <span m="1044220">edges</span> <span m="1044510">probably</span> <span m="1044880">here.</span> <span m="1046470">We</span> <span m="1046569">could</span> <span m="1046680">have</span> <span m="1046829">Penelope</span> <span m="1047420">here,</span> <span m="1049590">and</span> <span m="1050350">Katie</span> <span m="1053030">here.</span> <span m="1054200">And</span> <span m="1054650">well</span> <span m="1054790">probably</span> <span m="1055100">lots</span> <span m="1055390">more,</span> <span m="1055800">I</span> <span m="1055900">probably</span> <span m="1056210">don't</span> <span m="1056420">know</span> <span m="1056610">them</span> <span m="1056810">all.</span> <span m="1057400">And</span> <span m="1057830">Ben's</span> <span m="1058300">over</span> <span m="1058550">here</span> <span m="1059180">with</span> <span m="1059550">Nicole.</span> <span m="1061250">And</span> <span m="1061520">Nicole</span> <span m="1062190">got</span> <span m="1063280">Jude</span> <span m="1065130">and</span> <span m="1065940">Keith.</span> <span m="1068200">There's</span> <span m="1068380">actually</span> <span m="1068670">a</span> <span m="1068740">website</span> <span m="1069210">you</span> <span m="1069300">can</span> <span m="1069470">go</span> <span m="1069680">to</span> <span m="1071610">get</span> <span m="1071730">a</span> <span m="1071800">lot</span> <span m="1072010">of</span> <span m="1072060">these</span> <span m="1072470">things</span> <span m="1072810">here.</span> <span m="1073500">And</span> <span m="1073630">Katie</span> <span m="1074010">went</span> <span m="1074320">with</span> <span m="1074470">Josh.</span> <span m="1076830">It's</span> <span m="1076920">called</span> <span m="1077180">whosedatedwho.com,</span> <span m="1078060">and</span> <span m="1078790">you</span> <span m="1079070">get</span> <span m="1079220">big</span> <span m="1079440">graph,</span> <span m="1079840">you</span> <span m="1079930">could</span> <span m="1080040">start</span> <span m="1080300">filling</span> <span m="1080600">in</span> <span m="1080680">the</span> <span m="1080820">edges.</span> <span m="1081320">I</span> <span m="1081750">don't</span> <span m="1081820">know</span> <span m="1082020">how</span> <span m="1082295">reliable</span> <span m="1082570">it</span> <span m="1082900">is.</span> </p>
<p><span m="1083690">Now</span> <span m="1084280">it's</span> <span m="1084470">really</span> <span m="1084730">critical</span> <span m="1084990">that</span> <span m="1085250">we're</span> <span m="1085570">only</span> <span m="1085630">looking</span> <span m="1085930">edges</span> <span m="1086340">from</span> <span m="1086500">here</span> <span m="1086830">to</span> <span m="1086880">here.</span> <span m="1088270">All</span> <span m="1088400">right,</span> <span m="1088620">so</span> <span m="1088910">if</span> <span m="1089010">there's</span> <span m="1089190">an</span> <span m="1089260">edge</span> <span m="1089430">between</span> <span m="1089780">Tom</span> <span m="1090070">and</span> <span m="1090200">Ben,</span> <span m="1090650">I</span> <span m="1090820">don't</span> <span m="1090960">want</span> <span m="1091100">to</span> <span m="1091160">know</span> <span m="1091290">about</span> <span m="1091650">it.</span> <span m="1093740">Just</span> <span m="1094160">opposite-gender</span> <span m="1094840">partners.</span> <span m="1097540">OK</span> <span m="1097870">now</span> <span m="1098440">in</span> <span m="1098590">the</span> <span m="1098690">USA,</span> <span m="1101030">the</span> <span m="1101130">number</span> <span m="1101470">of</span> <span m="1101570">nodes</span> <span m="1102090">here</span> <span m="1104010">is</span> <span m="1104260">about</span> <span m="1104700">300</span> <span m="1105180">million.</span> <span m="1108640">About</span> <span m="1108830">three</span> <span m="1109060">million</span> <span m="1109330">people.</span> <span m="1110220">And</span> <span m="1110370">the</span> <span m="1110430">number</span> <span m="1110780">of</span> <span m="1110880">men</span> <span m="1111330">nodes,</span> <span m="1111990">male</span> <span m="1112290">nodes,</span> <span m="1113210">call</span> <span m="1113500">these</span> <span m="1113770">VM,</span> <span m="1115080">and</span> <span m="1115200">this</span> <span m="1115380">is</span> <span m="1115520">VW,</span> <span m="1116510">by</span> <span m="1117005">the</span> <span m="1117500">way,</span> <span m="1117730">I'm</span> <span m="1118090">using</span> <span m="1118415">cardinality</span> <span m="1118740">notation.</span> <span m="1119950">When</span> <span m="1120130">I</span> <span m="1120200">put</span> <span m="1121040">bars</span> <span m="1121580">around</span> <span m="1121950">a</span> <span m="1122000">set,</span> <span m="1122440">that</span> <span m="1122750">is</span> <span m="1123110">the</span> <span m="1123200">denoting</span> <span m="1123470">how</span> <span m="1123680">many</span> <span m="1124090">are</span> <span m="1124280">in</span> <span m="1124390">the</span> <span m="1124490">set.</span> </p>
<p><span m="1126050">In</span> <span m="1126170">the</span> <span m="1126280">US</span> <span m="1126790">there's</span> <span m="1126980">about</span> <span m="1128720">147.6</span> <span m="1130770">men</span> <span m="1131950">out</span> <span m="1132060">of</span> <span m="1132110">the</span> <span m="1132210">300.</span> <span m="1133360">And</span> <span m="1133680">the</span> <span m="1133740">number</span> <span m="1134080">of</span> <span m="1134160">women--</span> <span m="1135182">oh</span> <span m="1135644">we</span> <span m="1136106">got</span> <span m="1136568">a</span> <span m="1137030">w</span> <span m="1137390">here--</span> <span m="1138820">is</span> <span m="1139120">about</span> <span m="1139950">152.4</span> <span m="1141650">million.</span> <span m="1143240">So</span> <span m="1143420">there's</span> <span m="1143630">a</span> <span m="1143700">little</span> <span m="1143940">bit</span> <span m="1144130">more</span> <span m="1145340">nodes</span> <span m="1145740">on</span> <span m="1146000">this</span> <span m="1146280">side</span> <span m="1146490">of</span> <span m="1146540">the</span> <span m="1146620">graph,</span> <span m="1147120">than</span> <span m="1147220">that</span> <span m="1147440">side</span> <span m="1147940">in</span> <span m="1148070">the</span> <span m="1148150">US.</span> </p>
<p><span m="1150960">What</span> <span m="1151080">about</span> <span m="1151310">the</span> <span m="1151470">edges?</span> <span m="1152130">Any</span> <span m="1152350">idea</span> <span m="1152660">of</span> <span m="1152720">how</span> <span m="1152860">many</span> <span m="1153110">edges</span> <span m="1153490">there</span> <span m="1153710">are</span> <span m="1153850">here?</span> <span m="1157160">We</span> <span m="1157320">don't</span> <span m="1157520">know.</span> <span m="1157680">I</span> <span m="1157860">sure</span> <span m="1158120">as</span> <span m="1158220">heck</span> <span m="1158420">don't</span> <span m="1158650">know</span> <span m="1159430">how</span> <span m="1159620">many</span> <span m="1159850">edges</span> <span m="1160130">there</span> <span m="1160390">are.</span> <span m="1161070">So</span> <span m="1161420">that</span> <span m="1161640">we</span> <span m="1161730">don't</span> <span m="1161910">know.</span> <span m="1162410">The</span> <span m="1162520">cardinality</span> <span m="1162905">of</span> <span m="1163290">the</span> <span m="1163520">edge</span> <span m="1163690">set</span> <span m="1163790">we</span> <span m="1163970">don't</span> <span m="1164380">know,</span> <span m="1166160">and</span> <span m="1166340">we're</span> <span m="1166420">not</span> <span m="1166670">likely</span> <span m="1166970">to</span> <span m="1167040">figure</span> <span m="1167410">out.</span> <span m="1167940">I</span> <span m="1168070">don't</span> <span m="1168220">even</span> <span m="1168400">think</span> <span m="1168580">these</span> <span m="1168730">surveys,</span> <span m="1169300">really,</span> <span m="1170420">can</span> <span m="1170490">estimate</span> <span m="1170920">that.</span> <span m="1172900">But</span> <span m="1173090">what</span> <span m="1173340">we're</span> <span m="1173720">trying</span> <span m="1173950">to</span> <span m="1174010">figure</span> <span m="1174350">out</span> <span m="1175430">is</span> <span m="1175700">the</span> <span m="1175840">ratio</span> <span m="1178530">of</span> <span m="1178630">the</span> <span m="1178810">average</span> <span m="1179360">degree</span> <span m="1179860">of</span> <span m="1179950">the</span> <span m="1180040">men,</span> <span m="1180570">to</span> <span m="1182030">the</span> <span m="1182160">average</span> <span m="1182580">degree</span> <span m="1183000">of</span> <span m="1183060">the</span> <span m="1183170">women.</span> <span m="1184310">Because</span> <span m="1184620">the</span> <span m="1184710">number</span> <span m="1185040">of</span> <span m="1185130">opposite-gender</span> <span m="1185820">partners</span> <span m="1186280">you</span> <span m="1186400">have</span> <span m="1186730">is</span> <span m="1186820">your</span> <span m="1186970">degree</span> <span m="1187480">here,</span> <span m="1188420">and</span> <span m="1188530">you're</span> <span m="1188640">looking</span> <span m="1188870">for</span> <span m="1188950">the</span> <span m="1189100">average</span> <span m="1189480">guy</span> <span m="1189720">degree,</span> <span m="1190770">compared</span> <span m="1191150">to</span> <span m="1191200">the</span> <span m="1191360">average</span> <span m="1192730">female</span> <span m="1193120">degree</span> <span m="1193470">here.</span> <span m="1194370">That's</span> <span m="1194650">what</span> <span m="1194760">we're</span> <span m="1194950">after.</span> <span m="1195390">All</span> <span m="1195766">right</span> <span m="1196142">so</span> <span m="1196520">let's</span> <span m="1197090">find</span> <span m="1197400">that</span> <span m="1197570">quantity.</span> </p>
<p><span m="1200740">Let's</span> <span m="1201010">let</span> <span m="1201680">A</span> <span m="1202430">sub</span> <span m="1202690">m</span> <span m="1204010">equal</span> <span m="1204420">the</span> <span m="1204600">average</span> <span m="1208550">number</span> <span m="1210020">of</span> <span m="1210350">opposite-gender</span> <span m="1211120">partners</span> <span m="1220630">for</span> <span m="1220770">men.</span> <span m="1224830">And</span> <span m="1225150">we</span> <span m="1225240">can</span> <span m="1225370">let</span> <span m="1225590">A</span> <span m="1225800">W</span> <span m="1226290">be</span> <span m="1226800">the</span> <span m="1226900">same</span> <span m="1227210">thing</span> <span m="1227440">for</span> <span m="1227560">women.</span> </p>
<p></p>
<p><span m="1237310">All</span> <span m="1237560">right.</span> <span m="1238090">Now</span> <span m="1238480">we're</span> <span m="1238630">trying</span> <span m="1239040">to</span> <span m="1239100">figure</span> <span m="1239470">out</span> <span m="1240580">the</span> <span m="1240700">answer</span> <span m="1240950">to</span> <span m="1241050">this</span> <span m="1241130">question.</span> <span m="1242440">What</span> <span m="1242720">is</span> <span m="1244620">A</span> <span m="1244990">m,</span> <span m="1246030">the</span> <span m="1246190">average</span> <span m="1247110">guy</span> <span m="1247350">degree,</span> <span m="1248360">over</span> <span m="1248710">the</span> <span m="1248900">average</span> <span m="1250270">woman</span> <span m="1250550">degree.</span> <span m="1252550">And</span> <span m="1252720">in</span> <span m="1252780">particular,</span> <span m="1253260">the</span> <span m="1253430">University</span> <span m="1253900">of</span> <span m="1253950">Chicago</span> <span m="1254490">says,</span> <span m="1256080">they</span> <span m="1256250">say</span> <span m="1256650">it's</span> <span m="1256940">1.74.</span> <span m="1258480">That</span> <span m="1259130">the</span> <span m="1259350">average</span> <span m="1259670">guy</span> <span m="1259960">as</span> <span m="1260130">74%</span> <span m="1261020">more</span> <span m="1261730">opposite-gender</span> <span m="1262370">partners</span> <span m="1262770">than</span> <span m="1262860">the</span> <span m="1262980">average</span> <span m="1263210">woman.</span> <span m="1264550">ABC</span> <span m="1265010">News</span> <span m="1265520">says</span> <span m="1265840">it's</span> <span m="1266160">3.33,</span> <span m="1268140">that</span> <span m="1268560">is</span> <span m="1268810">233%</span> <span m="1270100">more</span> <span m="1270350">for</span> <span m="1270460">the</span> <span m="1270560">men,</span> <span m="1270830">than</span> <span m="1271000">the</span> <span m="1271240">women.</span> <span m="1272460">Now</span> <span m="1272590">we're</span> <span m="1272730">going</span> <span m="1272820">to</span> <span m="1272870">figure</span> <span m="1273110">this</span> <span m="1273320">out</span> <span m="1274460">what</span> <span m="1274630">this</span> <span m="1274800">ratio</span> <span m="1275210">is.</span> <span m="1276860">Just</span> <span m="1277070">use</span> <span m="1277210">a</span> <span m="1277270">little</span> <span m="1277530">bit</span> <span m="1277670">of</span> <span m="1277730">math</span> <span m="1278350">here,</span> <span m="1278540">and</span> <span m="1278600">a</span> <span m="1278670">little</span> <span m="1278880">bit</span> <span m="1279000">of</span> <span m="1279040">graph</span> <span m="1279340">theory.</span> </p>
<p><span m="1281260">So</span> <span m="1281640">let's</span> <span m="1281820">write</span> <span m="1282010">a</span> <span m="1282070">formula</span> <span m="1282900">for</span> <span m="1283110">A</span> <span m="1283380">m.</span> <span m="1287760">Well</span> <span m="1287950">we're</span> <span m="1288050">trying</span> <span m="1288450">to</span> <span m="1288520">figure</span> <span m="1288970">out</span> <span m="1290820">the</span> <span m="1291000">average</span> <span m="1291460">degree</span> <span m="1291900">over</span> <span m="1292140">here.</span> <span m="1294230">Well,</span> <span m="1294880">that's</span> <span m="1295120">pretty</span> <span m="1295320">simple.</span> <span m="1295660">We</span> <span m="1295770">just</span> <span m="1296000">add</span> <span m="1296400">up</span> <span m="1296610">all</span> <span m="1296930">the</span> <span m="1297020">degrees,</span> <span m="1298880">and</span> <span m="1299030">divide</span> <span m="1299370">by</span> <span m="1299460">the</span> <span m="1299580">number</span> <span m="1299810">of</span> <span m="1299880">nodes.</span> <span m="1300940">And</span> <span m="1301060">that'll</span> <span m="1301270">give</span> <span m="1301450">us</span> <span m="1301580">the</span> <span m="1301710">average</span> <span m="1301980">degree.</span> <span m="1304910">So</span> <span m="1305600">the</span> <span m="1305690">average</span> <span m="1306020">degree</span> <span m="1306320">is</span> <span m="1306460">the</span> <span m="1306580">sum</span> <span m="1307120">of</span> <span m="1307220">the</span> <span m="1307300">degrees,</span> <span m="1308390">over</span> <span m="1308630">all</span> <span m="1308830">men,</span> <span m="1309270">x</span> <span m="1309740">in</span> <span m="1309850">the</span> <span m="1309940">set</span> <span m="1310150">of</span> <span m="1310230">men,</span> <span m="1311670">of</span> <span m="1311810">the</span> <span m="1311890">degree</span> <span m="1312240">of</span> <span m="1312330">x,</span> <span m="1314590">divided</span> <span m="1315090">by</span> <span m="1315630">the</span> <span m="1315740">number</span> <span m="1316030">of</span> <span m="1316100">men.</span> <span m="1319830">Can</span> <span m="1320260">somebody</span> <span m="1320550">give</span> <span m="1320690">me</span> <span m="1320820">a</span> <span m="1320890">simpler</span> <span m="1321430">expression</span> <span m="1321990">for</span> <span m="1322090">this?</span> <span m="1324620">It</span> <span m="1324700">doesn't</span> <span m="1324980">have</span> <span m="1325150">that</span> <span m="1325330">nasty</span> <span m="1325680">sum</span> <span m="1326060">in</span> <span m="1326140">it?</span> </p>
<p><span m="1327454">AUDIENCE: E.</span> </p>
<p><span m="1329210">PROFESSOR: E.</span> <span m="1330045">The</span> <span m="1330300">cardinality</span> <span m="1331060">of</span> <span m="1331440">E.</span> <span m="1332440">I'm</span> <span m="1332730">adding</span> <span m="1333270">all</span> <span m="1333650">the</span> <span m="1333740">degrees</span> <span m="1334240">here.</span> <span m="1336960">Well</span> <span m="1337160">that's</span> <span m="1337440">just</span> <span m="1337640">another</span> <span m="1337910">way</span> <span m="1338020">of</span> <span m="1338120">counting</span> <span m="1338730">all</span> <span m="1339190">the</span> <span m="1339350">edges,</span> <span m="1340940">because</span> <span m="1341140">every</span> <span m="1341490">edge</span> <span m="1341820">shows</span> <span m="1342200">up</span> <span m="1342350">once,</span> <span m="1342670">and</span> <span m="1342750">only</span> <span m="1343010">once,</span> <span m="1344430">in</span> <span m="1344580">a</span> <span m="1344610">degree</span> <span m="1345270">count</span> <span m="1345520">here.</span> <span m="1346360">And</span> <span m="1346430">this</span> <span m="1346620">is</span> <span m="1346760">where,</span> <span m="1348150">we</span> <span m="1348290">use</span> <span m="1348490">the</span> <span m="1348590">fact</span> <span m="1348900">we</span> <span m="1348990">have</span> <span m="1349100">opposite-gender</span> <span m="1349950">partners.</span> <span m="1353020">Because</span> <span m="1354760">if</span> <span m="1354890">I</span> <span m="1354990">had</span> <span m="1355180">some</span> <span m="1355330">edges</span> <span m="1355590">over</span> <span m="1355780">here</span> <span m="1356130">they</span> <span m="1356250">wouldn't</span> <span m="1356540">get</span> <span m="1356700">counted</span> <span m="1357320">in</span> <span m="1357440">sum</span> <span m="1357720">of</span> <span m="1357760">the</span> <span m="1357840">degrees</span> <span m="1358210">here.</span> <span m="1360240">All</span> <span m="1360320">right</span> <span m="1360800">so</span> <span m="1361820">this</span> <span m="1362080">is</span> <span m="1362230">just</span> <span m="1364760">the</span> <span m="1364880">cardinality</span> <span m="1365255">of</span> <span m="1365630">the</span> <span m="1365970">number</span> <span m="1366230">of</span> <span m="1366300">edges,</span> <span m="1366580">divided</span> <span m="1366970">by</span> <span m="1367170">the</span> <span m="1367270">number</span> <span m="1367540">of</span> <span m="1367590">men.</span> <span m="1368970">Any</span> <span m="1369240">questions</span> <span m="1369700">about</span> <span m="1369950">that?</span> <span m="1370380">Because</span> <span m="1370570">this</span> <span m="1370690">is</span> <span m="1370760">an</span> <span m="1371123">important</span> <span m="1372370">statement</span> <span m="1372780">about</span> <span m="1373020">graphs</span> <span m="1373450">in</span> <span m="1373550">general.</span> </p>
<p><span m="1374930">When</span> <span m="1375060">I</span> <span m="1375110">have</span> <span m="1375270">a</span> <span m="1375320">graph</span> <span m="1375700">like</span> <span m="1376000">this--</span> <span m="1377390">which</span> <span m="1377500">is</span> <span m="1377610">called</span> <span m="1377790">a</span> <span m="1377830">bipartite</span> <span m="1378380">graph,</span> <span m="1378710">we'll</span> <span m="1378800">talk</span> <span m="1378990">about</span> <span m="1379210">more</span> <span m="1379550">in</span> <span m="1379790">a</span> <span m="1380150">little</span> <span m="1380530">bit.</span> <span m="1380670">But</span> <span m="1380800">where</span> <span m="1381040">the</span> <span m="1381210">edges</span> <span m="1381310">go</span> <span m="1381410">from</span> <span m="1381660">the</span> <span m="1381720">left</span> <span m="1381830">to</span> <span m="1382100">the</span> <span m="1382210">right</span> <span m="1382290">if</span> <span m="1382630">I</span> <span m="1382720">sum</span> <span m="1382995">the</span> <span m="1383270">degrees</span> <span m="1383360">on</span> <span m="1383450">the</span> <span m="1383710">left,</span> <span m="1384210">I'm</span> <span m="1384350">just</span> <span m="1384520">counting</span> <span m="1384820">the</span> <span m="1384880">number</span> <span m="1385070">of</span> <span m="1385150">edges.</span> <span m="1389060">All</span> <span m="1389280">right,</span> <span m="1390900">let's</span> <span m="1391160">figure</span> <span m="1391400">out</span> <span m="1391490">a</span> <span m="1391540">formula</span> <span m="1392070">for</span> <span m="1392210">the</span> <span m="1393020">average</span> <span m="1393490">number</span> <span m="1393840">of</span> <span m="1393900">partners</span> <span m="1394350">for</span> <span m="1394430">the</span> <span m="1394530">women.</span> <span m="1396340">That</span> <span m="1396550">simple</span> <span m="1397020">that's</span> <span m="1397170">just</span> <span m="1399070">sum</span> <span m="1399580">x</span> <span m="1400040">over</span> <span m="1400340">the</span> <span m="1400590">women.</span> <span m="1403240">The</span> <span m="1403310">degree</span> <span m="1403700">of</span> <span m="1403790">x,</span> <span m="1406520">divided</span> <span m="1406930">by</span> <span m="1407160">the</span> <span m="1407280">number</span> <span m="1407510">of</span> <span m="1407570">women.</span> <span m="1410900">Let</span> <span m="1410960">me</span> <span m="1411080">rewrite</span> <span m="1411470">that</span> <span m="1411690">so</span> <span m="1411850">it's</span> <span m="1412100">clearer.</span> <span m="1416150">What's</span> <span m="1416450">a</span> <span m="1416520">simpler</span> <span m="1416940">expression</span> <span m="1417410">for</span> <span m="1417500">this?</span> </p>
<p><span m="1418922">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="1423080">PROFESSOR: Yeah,</span> <span m="1424210">this</span> <span m="1424570">sum,</span> <span m="1425920">adding</span> <span m="1426310">the</span> <span m="1426380">degrees</span> <span m="1426830">of</span> <span m="1426900">the</span> <span m="1427000">women,</span> <span m="1428970">is</span> <span m="1429200">just</span> <span m="1430630">the</span> <span m="1430740">number</span> <span m="1430990">of</span> <span m="1431080">edges,</span> <span m="1432410">right.</span> <span m="1434690">So</span> <span m="1435160">that</span> <span m="1435450">is</span> <span m="1436700">cardinality</span> <span m="1437290">of</span> <span m="1437350">edges,</span> <span m="1437950">divided</span> <span m="1438340">by</span> <span m="1438450">the</span> <span m="1438560">number</span> <span m="1438850">of</span> <span m="1438940">women.</span> <span m="1440280">All</span> <span m="1440670">right,</span> <span m="1442450">well</span> <span m="1443360">now</span> <span m="1443650">we</span> <span m="1443760">can</span> <span m="1443900">write,</span> <span m="1444740">solve</span> <span m="1445080">for</span> <span m="1445220">our</span> <span m="1445310">formula,</span> <span m="1446700">average</span> <span m="1447070">over</span> <span m="1447220">men</span> <span m="1449490">over</span> <span m="1450130">average</span> <span m="1450530">of</span> <span m="1450590">the</span> <span m="1450690">women.</span> <span m="1452320">That's</span> <span m="1454356">E</span> <span m="1455660">over</span> <span m="1456050">VM,</span> <span m="1458130">divided</span> <span m="1458600">by</span> <span m="1459110">E</span> <span m="1460190">over</span> <span m="1460680">VW.</span> <span m="1462126">Wow,</span> <span m="1464060">this</span> <span m="1464240">is</span> <span m="1464360">nice.</span> <span m="1464900">I</span> <span m="1465050">don't</span> <span m="1465250">know</span> <span m="1465530">the</span> <span m="1465630">number</span> <span m="1465770">of</span> <span m="1465810">edges</span> <span m="1466030">is,</span> <span m="1467430">but</span> <span m="1467560">it</span> <span m="1467640">just</span> <span m="1467840">canceled</span> <span m="1468260">out.</span> <span m="1470290">And</span> <span m="1470440">this</span> <span m="1470620">is</span> <span m="1470750">just</span> <span m="1471470">the</span> <span m="1471570">number</span> <span m="1471860">of</span> <span m="1471940">women,</span> <span m="1473300">divided</span> <span m="1473750">by</span> <span m="1473880">the</span> <span m="1473990">number</span> <span m="1474330">of</span> <span m="1474400">men.</span> </p>
<p><span m="1477430">And</span> <span m="1477560">in</span> <span m="1477620">fact</span> <span m="1477900">we</span> <span m="1478040">know</span> <span m="1478340">that.</span> <span m="1480740">That's</span> <span m="1481070">this</span> <span m="1481370">number,</span> <span m="1481610">divided</span> <span m="1481940">by</span> <span m="1482060">that</span> <span m="1482290">number,</span> <span m="1482660">which</span> <span m="1482980">is</span> <span m="1483170">about</span> <span m="1486040">1.0325.</span> <span m="1492090">So</span> <span m="1492270">we</span> <span m="1492440">just</span> <span m="1492700">proved,</span> <span m="1494740">that</span> <span m="1494900">on</span> <span m="1495090">average,</span> <span m="1496940">a</span> <span m="1497030">man</span> <span m="1497550">has</span> <span m="1498410">3%,</span> <span m="1499330">or</span> <span m="1499430">3</span> <span m="1499660">and</span> <span m="1499900">1/4%</span> <span m="1500540">more</span> <span m="1500830">opposite-gender</span> <span m="1501540">partners</span> <span m="1501950">than</span> <span m="1502080">women.</span> <span m="1505010">No</span> <span m="1505240">need</span> <span m="1505510">to</span> <span m="1505580">do</span> <span m="1505710">the</span> <span m="1505870">interviews,</span> <span m="1507030">or</span> <span m="1507170">spend</span> <span m="1507440">years</span> <span m="1507810">doing.</span> <span m="1508800">That</span> <span m="1509050">is</span> <span m="1509240">the</span> <span m="1509390">answer.</span> <span m="1510390">And</span> <span m="1510580">it</span> <span m="1510630">has</span> <span m="1510850">nothing</span> <span m="1511250">to</span> <span m="1511350">do</span> <span m="1511800">with</span> <span m="1511950">the</span> <span m="1512110">promiscuity</span> <span m="1512880">of</span> <span m="1512960">men,</span> <span m="1514120">or</span> <span m="1514340">women,</span> <span m="1515480">nothing</span> <span m="1515780">at</span> <span m="1515860">all.</span> </p>
<p><span m="1519080">So</span> <span m="1519880">the</span> <span m="1519980">Chicago</span> <span m="1520470">study</span> <span m="1521190">is</span> <span m="1521430">way</span> <span m="1521690">off,</span> <span m="1522090">and</span> <span m="1522240">the</span> <span m="1522300">ABC</span> <span m="1522730">New</span> <span m="1522990">study</span> <span m="1523300">is</span> <span m="1523410">completely</span> <span m="1523810">nuts.</span> <span m="1525400">It</span> <span m="1525490">just</span> <span m="1525680">can't</span> <span m="1525990">be</span> <span m="1526120">right,</span> <span m="1527300">this</span> <span m="1527500">is</span> <span m="1527600">a</span> <span m="1527670">proof.</span> <span m="1530050">Now</span> <span m="1530960">what</span> <span m="1531160">happened</span> <span m="1531650">here?</span> <span m="1533740">Well</span> <span m="1534180">what's</span> <span m="1534600">going</span> <span m="1534810">on,</span> <span m="1534890">what's</span> <span m="1535090">the</span> <span m="1535200">reason</span> <span m="1535610">for</span> <span m="1535730">why</span> <span m="1535980">this</span> <span m="1536230">is</span> <span m="1536460">true?</span> <span m="1538110">Yeah?</span> </p>
<p><span m="1538410">AUDIENCE: A</span> <span m="1538877">male</span> <span m="1539344">has</span> <span m="1539811">a</span> <span m="1540278">female</span> <span m="1540745">partner</span> <span m="1541212">then</span> <span m="1541679">the</span> <span m="1542146">female</span> <span m="1542613">has</span> <span m="1543080">a</span> <span m="1543550">male</span> <span m="1543970">partner.</span> </p>
<p><span m="1544320">PROFESSOR: Yeah.</span> </p>
<p><span m="1544670">AUDIENCE: You're</span> <span m="1545020">not</span> <span m="1545220">looking</span> <span m="1545600">at</span> <span m="1546025">like</span> <span m="1546450">how</span> <span m="1546710">many</span> <span m="1546990">males</span> <span m="1547456">are</span> <span m="1547922">going</span> <span m="1548388">to</span> <span m="1548854">one</span> <span m="1549320">female.</span> <span m="1549786">The</span> <span m="1550252">promiscuity</span> <span m="1550718">isn't</span> <span m="1551184">even</span> <span m="1551650">a</span> <span m="1552116">part</span> <span m="1552590">of</span> <span m="1552840">the</span> <span m="1553140">question.</span> </p>
<p><span m="1553260">PROFESSOR: That's</span> <span m="1553710">right.</span> <span m="1554110">It</span> <span m="1554380">takes</span> <span m="1554590">two</span> <span m="1554730">to</span> <span m="1554850">tango.</span> <span m="1556610">Every</span> <span m="1556800">time</span> <span m="1557000">you</span> <span m="1557080">got</span> <span m="1557240">a</span> <span m="1557280">guy,</span> <span m="1557580">you</span> <span m="1557860">got</span> <span m="1557910">a</span> <span m="1558255">women.</span> <span m="1558880">And</span> <span m="1559290">you</span> <span m="1559380">have</span> <span m="1559470">the</span> <span m="1559560">number</span> <span m="1560000">of</span> <span m="1560300">relationships</span> <span m="1561060">going.</span> <span m="1561920">The</span> <span m="1562080">average</span> <span m="1562400">for</span> <span m="1562480">the</span> <span m="1562580">men</span> <span m="1562820">is</span> <span m="1562940">that</span> <span m="1563130">number,</span> <span m="1563430">divided</span> <span m="1563820">by</span> <span m="1563930">the</span> <span m="1564020">men.</span> <span m="1564530">Average</span> <span m="1564895">for</span> <span m="1565260">the</span> <span m="1565480">women</span> <span m="1565620">is</span> <span m="1565900">that</span> <span m="1566070">same</span> <span m="1566480">number,</span> <span m="1566810">divided</span> <span m="1567170">by</span> <span m="1567270">the</span> <span m="1567390">women.</span> <span m="1567810">And</span> <span m="1568140">so</span> <span m="1568270">if</span> <span m="1568380">there's</span> <span m="1568590">more</span> <span m="1568830">women,</span> <span m="1569130">they're</span> <span m="1569240">going</span> <span m="1569340">to</span> <span m="1569390">have</span> <span m="1569570">less</span> <span m="1569770">partners</span> <span m="1570130">on</span> <span m="1570260">average.</span> <span m="1571080">Has</span> <span m="1571420">to</span> <span m="1571520">be.</span> <span m="1572670">So</span> <span m="1572760">it</span> <span m="1572830">really</span> <span m="1573140">was</span> <span m="1573270">a</span> <span m="1573620">stupid</span> <span m="1574010">question.</span> <span m="1574420">It's</span> <span m="1574530">very,</span> <span m="1574860">very</span> <span m="1575000">simple</span> <span m="1575340">to</span> <span m="1575450">answer.</span> </p>
<p><span m="1577970">Now</span> <span m="1578190">as</span> <span m="1578320">it</span> <span m="1578430">turns</span> <span m="1578820">out</span> <span m="1579040">there</span> <span m="1579240">are</span> <span m="1579490">endless</span> <span m="1580010">studies</span> <span m="1580420">like</span> <span m="1580690">this,</span> <span m="1581340">in</span> <span m="1581500">the</span> <span m="1581590">literature.</span> <span m="1583090">In</span> <span m="1583190">fact,</span> <span m="1583590">a</span> <span m="1583690">few</span> <span m="1583740">years</span> <span m="1584050">ago</span> <span m="1584260">the</span> <span m="1584360">Boston</span> <span m="1584860">Globe</span> <span m="1585940">ran</span> <span m="1586190">an</span> <span m="1586270">explosive</span> <span m="1586980">story</span> <span m="1588280">about</span> <span m="1588910">the</span> <span m="1589030">study</span> <span m="1589450">habits</span> <span m="1590440">of</span> <span m="1590580">students</span> <span m="1591140">on</span> <span m="1591270">Boston-area</span> <span m="1592000">campuses.</span> <span m="1593530">And</span> <span m="1593680">their</span> <span m="1593920">surveys</span> <span m="1594550">show</span> <span m="1594960">that,</span> <span m="1595230">on</span> <span m="1595520">average,</span> <span m="1596340">minority</span> <span m="1596930">students</span> <span m="1598320">tended</span> <span m="1598790">to</span> <span m="1598890">study</span> <span m="1599270">with</span> <span m="1599430">non-minority</span> <span m="1600300">students</span> <span m="1601380">more</span> <span m="1601730">than</span> <span m="1601850">the</span> <span m="1602000">other</span> <span m="1602180">way</span> <span m="1602350">around.</span> </p>
<p><span m="1606350">And</span> <span m="1606470">they</span> <span m="1606590">want</span> <span m="1606810">on</span> <span m="1607030">great</span> <span m="1607350">length</span> <span m="1608010">consulting</span> <span m="1608380">the</span> <span m="1608750">experts</span> <span m="1609250">as</span> <span m="1609350">to</span> <span m="1609470">why</span> <span m="1609930">this</span> <span m="1610170">might</span> <span m="1610430">be</span> <span m="1610550">true.</span> <span m="1610980">Why</span> <span m="1611170">is</span> <span m="1611330">it</span> <span m="1611460">the</span> <span m="1611550">minority</span> <span m="1612000">students</span> <span m="1612750">study</span> <span m="1613230">with</span> <span m="1613750">non-minority</span> <span m="1614380">students</span> <span m="1614900">more</span> <span m="1615910">than</span> <span m="1616030">the</span> <span m="1616150">other</span> <span m="1616310">way</span> <span m="1616460">around.</span> <span m="1616870">Now</span> <span m="1617010">can</span> <span m="1617160">anyone</span> <span m="1617560">tell</span> <span m="1617820">me</span> <span m="1617960">why</span> <span m="1618180">it</span> <span m="1618250">is</span> <span m="1618390">certainly</span> <span m="1618940">true,</span> <span m="1619750">and</span> <span m="1619980">not</span> <span m="1620260">surprising,</span> <span m="1621400">why</span> <span m="1621600">that's</span> <span m="1621830">the</span> <span m="1621930">case?</span> </p>
<p><span m="1622205">AUDIENCE: Because</span> <span m="1622480">they're</span> <span m="1622890">the</span> <span m="1623300">minority.</span> </p>
<p><span m="1624120">PROFESSOR: Because</span> <span m="1624430">they're</span> <span m="1624670">a</span> <span m="1624790">minority.</span> <span m="1625190">There's</span> <span m="1625410">fewer</span> <span m="1626290">minorities</span> <span m="1626920">than</span> <span m="1627070">non-minorities.</span> <span m="1627745">End</span> <span m="1628130">of</span> <span m="1629020">story,</span> <span m="1629400">we</span> <span m="1629530">don't</span> <span m="1629770">need</span> <span m="1630020">this</span> <span m="1630240">sociology</span> <span m="1631470">PhD</span> <span m="1632030">from</span> <span m="1632160">down</span> <span m="1632390">the</span> <span m="1632470">street</span> <span m="1632760">to</span> <span m="1632860">explain</span> <span m="1633220">it</span> <span m="1633320">to</span> <span m="1633460">us.</span> </p>
<p><span m="1636780">We're</span> <span m="1636950">going</span> <span m="1637040">to</span> <span m="1637100">see</span> <span m="1637230">a</span> <span m="1637280">lot</span> <span m="1637470">of</span> <span m="1637550">other</span> <span m="1637690">bogus</span> <span m="1638110">studies</span> <span m="1638570">later.</span> <span m="1639060">This</span> <span m="1639270">is</span> <span m="1639580">not</span> <span m="1639840">unusual,</span> <span m="1640780">especially</span> <span m="1641350">when</span> <span m="1641430">we</span> <span m="1641500">get</span> <span m="1641630">the</span> <span m="1641700">probability.</span> <span m="1642270">Just</span> <span m="1642640">every</span> <span m="1642880">day</span> <span m="1643080">there's</span> <span m="1643280">a</span> <span m="1643340">new</span> <span m="1643520">one</span> <span m="1644150">in</span> <span m="1644270">probability.</span> <span m="1646410">Any</span> <span m="1646610">questions</span> <span m="1647090">about</span> <span m="1647360">this</span> <span m="1647610">before</span> <span m="1647880">we</span> <span m="1648010">leave?</span> <span m="1649310">Unfortunately</span> <span m="1649810">that's</span> <span m="1650090">most</span> <span m="1650370">all</span> <span m="1650580">we'll</span> <span m="1650750">say</span> <span m="1651000">about</span> <span m="1651270">sex</span> <span m="1651670">today.</span> </p>
<p><span m="1655590">OK.</span> <span m="1656800">But</span> <span m="1657490">now,</span> <span m="1658220">in</span> <span m="1658350">this</span> <span m="1658540">example,</span> <span m="1659680">we</span> <span m="1659860">used</span> <span m="1660810">an</span> <span m="1661060">edge</span> <span m="1661410">in</span> <span m="1661500">the</span> <span m="1661590">graph</span> <span m="1662110">to</span> <span m="1662230">denote</span> <span m="1662710">some</span> <span m="1662970">kind</span> <span m="1663180">of</span> <span m="1663270">affinity</span> <span m="1664790">between</span> <span m="1665480">two</span> <span m="1665640">nodes.</span> <span m="1666530">The</span> <span m="1666660">two</span> <span m="1666810">nodes</span> <span m="1667160">liked</span> <span m="1667550">each</span> <span m="1667740">other</span> <span m="1667950">in</span> <span m="1668040">some</span> <span m="1668290">sense</span> <span m="1668660">if</span> <span m="1668750">they</span> <span m="1668870">were</span> <span m="1668980">connected</span> <span m="1669410">by</span> <span m="1669610">an</span> <span m="1669720">edge,</span> <span m="1670010">or</span> <span m="1670090">they</span> <span m="1670260">had</span> <span m="1670310">a</span> <span m="1670630">relationship</span> <span m="1670950">of</span> <span m="1671050">some</span> <span m="1671270">kind.</span> <span m="1672520">There's</span> <span m="1672730">lots</span> <span m="1673050">of</span> <span m="1673140">examples</span> <span m="1673590">in</span> <span m="1674040">computer</span> <span m="1674390">science</span> <span m="1674490">where</span> <span m="1674860">you</span> <span m="1675350">use</span> <span m="1675780">an</span> <span m="1675880">edge</span> <span m="1676240">to</span> <span m="1676370">denote</span> <span m="1676610">just</span> <span m="1676930">the</span> <span m="1677080">opposite.</span> <span m="1677453">That</span> <span m="1678200">the</span> <span m="1678420">two</span> <span m="1678610">nodes</span> <span m="1679090">can't</span> <span m="1679400">be</span> <span m="1679510">near</span> <span m="1679770">each</span> <span m="1679920">other,</span> <span m="1680200">or</span> <span m="1680330">don't</span> <span m="1680570">like</span> <span m="1680800">each</span> <span m="1680970">other.</span> </p>
<p><span m="1682750">For</span> <span m="1682900">example,</span> <span m="1683500">consider</span> <span m="1683850">the</span> <span m="1683970">problem</span> <span m="1684420">of</span> <span m="1684500">scheduling</span> <span m="1685110">final</span> <span m="1685470">exams</span> <span m="1686550">at</span> <span m="1686680">MIT.</span> <span m="1688080">And</span> <span m="1688170">they</span> <span m="1688280">do</span> <span m="1688490">this</span> <span m="1688940">after</span> <span m="1689250">they</span> <span m="1689380">find</span> <span m="1689660">out</span> <span m="1689770">all</span> <span m="1689930">of</span> <span m="1690010">your</span> <span m="1690150">schedules,</span> <span m="1691310">and</span> <span m="1691490">they</span> <span m="1691580">try</span> <span m="1691760">to</span> <span m="1691860">schedule</span> <span m="1692350">the</span> <span m="1692480">exams</span> <span m="1693850">so</span> <span m="1694010">that</span> <span m="1694350">you</span> <span m="1694430">don't</span> <span m="1694560">have</span> <span m="1694640">to</span> <span m="1694740">take</span> <span m="1694930">two</span> <span m="1695110">at</span> <span m="1695190">once,</span> <span m="1695650">or</span> <span m="1695750">there's</span> <span m="1696020">as</span> <span m="1696060">little</span> <span m="1696270">of</span> <span m="1696540">that</span> <span m="1696930">as</span> <span m="1697090">possible.</span> <span m="1699270">For</span> <span m="1699330">example,</span> <span m="1699940">let's</span> <span m="1700120">do</span> <span m="1701820">an</span> <span m="1701890">example</span> <span m="1702360">here.</span> </p>
<p><span m="1713460">Say</span> <span m="1713700">we</span> <span m="1713830">look</span> <span m="1714030">at</span> <span m="1714510">these</span> <span m="1714830">five</span> <span m="1715350">classes.</span> <span m="1724590">Take</span> <span m="1724830">6041.</span> <span m="1727220">And</span> <span m="1727300">this</span> <span m="1727470">may</span> <span m="1727600">not</span> <span m="1727810">be</span> <span m="1728040">totally</span> <span m="1728510">accurate,</span> <span m="1729070">but</span> <span m="1729660">roughly.</span> <span m="1737250">So</span> <span m="1737400">I've</span> <span m="1737500">got</span> <span m="1737870">five</span> <span m="1738240">MIT</span> <span m="1738640">classes,</span> <span m="1739670">and</span> <span m="1739850">I'm</span> <span m="1739950">going</span> <span m="1740060">to</span> <span m="1740120">put</span> <span m="1740320">an</span> <span m="1740480">edge</span> <span m="1740870">between</span> <span m="1741830">pairs</span> <span m="1742190">of</span> <span m="1742290">classes</span> <span m="1743120">that</span> <span m="1743630">have</span> <span m="1743920">overlapping</span> <span m="1744550">student</span> <span m="1744940">enrollment.</span> </p>
<p><span m="1748630">So</span> <span m="1748940">in</span> <span m="1749010">this</span> <span m="1749200">case,</span> <span m="1749560">for</span> <span m="1749660">example,</span> <span m="1751200">we've</span> <span m="1751490">assumed</span> <span m="1751770">in</span> <span m="1752050">the</span> <span m="1752180">drawing</span> <span m="1752460">of</span> <span m="1752650">his</span> <span m="1752905">graph,</span> <span m="1753160">that</span> <span m="1753260">you</span> <span m="1753390">can't</span> <span m="1753750">have</span> <span m="1754390">our</span> <span m="1754660">exam</span> <span m="1755080">the</span> <span m="1755160">same</span> <span m="1755420">time</span> <span m="1755630">is</span> <span m="1755710">6002,</span> <span m="1757280">on</span> <span m="1757440">the</span> <span m="1757540">assumption</span> <span m="1758010">there's</span> <span m="1758150">students</span> <span m="1758810">in</span> <span m="1758940">both</span> <span m="1759170">classes.</span> <span m="1760640">But</span> <span m="1760880">you</span> <span m="1761000">could</span> <span m="1761330">have</span> <span m="1761560">our</span> <span m="1761720">exam</span> <span m="1762160">the</span> <span m="1762250">same</span> <span m="1762540">time</span> <span m="1762780">as</span> <span m="1762880">6034.</span> <span m="1764560">Because</span> <span m="1765380">there's</span> <span m="1765540">not</span> <span m="1765920">an</span> <span m="1766245">overlapping</span> <span m="1767490">student</span> <span m="1767730">in</span> <span m="1767810">both</span> <span m="1768010">classes,</span> <span m="1768510">so</span> <span m="1768600">the</span> <span m="1768680">exams</span> <span m="1769110">could</span> <span m="1769350">be</span> <span m="1769450">scheduled</span> <span m="1769850">at</span> <span m="1769910">the</span> <span m="1770020">same</span> <span m="1770280">time.</span> <span m="1771510">So</span> <span m="1771540">we've</span> <span m="1771650">used</span> <span m="1772000">a</span> <span m="1772040">graph</span> <span m="1772580">to</span> <span m="1772730">represent</span> <span m="1773720">which</span> <span m="1774300">courses</span> <span m="1775280">can't</span> <span m="1775730">have</span> <span m="1775930">their</span> <span m="1776030">exam</span> <span m="1776400">at</span> <span m="1776480">the</span> <span m="1776730">same</span> <span m="1777020">time.</span> </p>
<p><span m="1778740">Now</span> <span m="1778920">let's</span> <span m="1779120">also</span> <span m="1779380">suppose</span> <span m="1779890">we</span> <span m="1780000">have</span> <span m="1780210">a</span> <span m="1780560">set</span> <span m="1780770">of</span> <span m="1780860">slots</span> <span m="1781390">for</span> <span m="1781460">the</span> <span m="1781600">exam.</span> <span m="1785360">And</span> <span m="1785740">say</span> <span m="1785900">they're</span> <span m="1786080">all</span> <span m="1786220">on</span> <span m="1786360">a</span> <span m="1786420">Wednesday.</span> <span m="1787670">And</span> <span m="1787790">the</span> <span m="1787870">first</span> <span m="1788160">slot</span> <span m="1788400">is</span> <span m="1788510">Wednesday</span> <span m="1789050">from</span> <span m="1789240">5:00</span> <span m="1789580">to</span> <span m="1789680">7:00.</span> <span m="1791480">And</span> <span m="1791680">the</span> <span m="1791760">next</span> <span m="1792050">one</span> <span m="1792290">is</span> <span m="1792860">7:00</span> <span m="1793300">to</span> <span m="1793420">9:00.</span> <span m="1795430">And</span> <span m="1795600">then,</span> <span m="1796360">the</span> <span m="1796440">next</span> <span m="1796740">one</span> <span m="1796890">is</span> <span m="1797010">9:00</span> <span m="1797440">to</span> <span m="1797610">11:00.</span> <span m="1800070">And</span> <span m="1800330">then</span> <span m="1800480">11:00</span> <span m="1800980">to</span> <span m="1801120">1:00</span> <span m="1801740">in</span> <span m="1801820">the</span> <span m="1801910">morning,</span> <span m="1803230">and</span> <span m="1803460">then</span> <span m="1803620">1:00</span> <span m="1803880">to</span> <span m="1803970">3:00,</span> <span m="1805290">getting</span> <span m="1805740">pretty</span> <span m="1806080">late.</span> <span m="1807930">And</span> <span m="1809320">your</span> <span m="1809490">job</span> <span m="1809930">is</span> <span m="1810090">to</span> <span m="1810220">figure</span> <span m="1810560">out</span> <span m="1812330">how</span> <span m="1812540">not</span> <span m="1812920">to</span> <span m="1812980">have</span> <span m="1813230">to</span> <span m="1813320">use</span> <span m="1813750">these</span> <span m="1813970">later</span> <span m="1814260">exam</span> <span m="1814660">slots.</span> <span m="1815290">You'd</span> <span m="1815460">like</span> <span m="1815640">to</span> <span m="1815720">use</span> <span m="1815970">as</span> <span m="1816120">few</span> <span m="1816440">as</span> <span m="1816580">possible</span> <span m="1817140">so</span> <span m="1817270">you're</span> <span m="1817360">not</span> <span m="1818880">going</span> <span m="1819270">too</span> <span m="1819410">late</span> <span m="1819770">night,</span> <span m="1820180">or</span> <span m="1820930">come</span> <span m="1821590">before</span> <span m="1821880">the</span> <span m="1821970">holidays,</span> <span m="1822590">so</span> <span m="1822670">you're</span> <span m="1822770">not</span> <span m="1822930">having</span> <span m="1823140">exams</span> <span m="1823580">on</span> <span m="1823790">Christmas</span> <span m="1824210">and</span> <span m="1824290">New</span> <span m="1824420">Year's,</span> <span m="1824920">for</span> <span m="1825080">example.</span> </p>
<p><span m="1828450">So</span> <span m="1830030">the</span> <span m="1830170">goal</span> <span m="1830620">is</span> <span m="1830770">to</span> <span m="1830890">assign</span> <span m="1831440">slots</span> <span m="1833060">to</span> <span m="1833210">the</span> <span m="1833330">nodes.</span> <span m="1833840">Put</span> <span m="1833990">every</span> <span m="1834210">node</span> <span m="1835070">in</span> <span m="1835210">a</span> <span m="1835270">slot</span> <span m="1836290">so</span> <span m="1836460">you</span> <span m="1836620">don't</span> <span m="1837090">have</span> <span m="1839070">nodes</span> <span m="1839325">hooked</span> <span m="1839580">by</span> <span m="1839760">an</span> <span m="1840010">edge</span> <span m="1840300">getting</span> <span m="1840540">the</span> <span m="1840630">same</span> <span m="1840930">slot.</span> <span m="1842110">Now</span> <span m="1842420">this</span> <span m="1842680">is</span> <span m="1842810">an</span> <span m="1842900">example</span> <span m="1843770">of</span> <span m="1844010">what's</span> <span m="1844190">called</span> <span m="1844400">a</span> <span m="1844440">graph</span> <span m="1844940">coloring</span> <span m="1845490">problem.</span> <span m="1846590">So</span> <span m="1846730">let's</span> <span m="1846910">define</span> <span m="1847280">that.</span> </p>
<p></p>
<p><span m="1877680">Given</span> <span m="1877970">a</span> <span m="1878060">graph</span> <span m="1878410">G,</span> <span m="1883220">and</span> <span m="1884300">K</span> <span m="1884740">colors,</span> <span m="1892590">assign</span> <span m="1894680">a</span> <span m="1894790">color</span> <span m="1897520">to</span> <span m="1897780">each</span> <span m="1898110">node,</span> <span m="1904020">so</span> <span m="1904180">that</span> <span m="1904830">adjacent</span> <span m="1905630">nodes</span> <span m="1914420">get</span> <span m="1915200">different</span> <span m="1915660">colors.</span> <span m="1923154">All</span> <span m="1923650">right,</span> <span m="1924130">and</span> <span m="1924310">then</span> <span m="1925310">the</span> <span m="1925470">minimum</span> <span m="1926040">number</span> <span m="1926330">of</span> <span m="1926390">colors</span> <span m="1926780">you</span> <span m="1926920">need</span> <span m="1928120">is</span> <span m="1928280">called</span> <span m="1928600">the</span> <span m="1928690">chromatic</span> <span m="1929310">number</span> <span m="1929580">of</span> <span m="1929690">the</span> <span m="1930080">graph.</span> <span m="1931750">So</span> <span m="1931870">the</span> <span m="1932000">minimum</span> <span m="1938270">value</span> <span m="1938770">of</span> <span m="1938870">K,</span> <span m="1942190">for</span> <span m="1942560">which</span> <span m="1944970">such</span> <span m="1945570">a</span> <span m="1945710">coloring</span> <span m="1946190">exist,</span> <span m="1956460">is</span> <span m="1956820">the</span> <span m="1957370">chromatic</span> <span m="1958320">number</span> <span m="1966880">OF</span> <span m="1966970">the</span> <span m="1967090">graph.</span> <span m="1968840">And</span> <span m="1969020">it's</span> <span m="1969160">denoted</span> <span m="1969660">by</span> <span m="1970390">this</span> <span m="1970610">symbol</span> <span m="1970940">chi</span> <span m="1971265">of</span> <span m="1971900">G.</span> <span m="1980110">Because</span> <span m="1980310">usually</span> <span m="1980740">you</span> <span m="1980830">want</span> <span m="1980990">to</span> <span m="1981030">use</span> <span m="1981200">a</span> <span m="1981250">small</span> <span m="1981550">number</span> <span m="1981760">of</span> <span m="1981850">colors.</span> </p>
<p><span m="1982330">Now</span> <span m="1983850">what</span> <span m="1984140">does</span> <span m="1984290">a</span> <span m="1984430">color</span> <span m="1984940">represent</span> <span m="1986360">when</span> <span m="1986530">we're</span> <span m="1987270">dealing</span> <span m="1987620">with</span> <span m="1987740">this</span> <span m="1987930">problem?</span> <span m="1989220">What's</span> <span m="1989330">the</span> <span m="1989420">meaning</span> <span m="1989790">of</span> <span m="1989870">a</span> <span m="1989940">color?</span> </p>
<p><span m="1990760">AUDIENCE: Time</span> <span m="1991140">slot.</span> </p>
<p><span m="1991900">PROFESSOR: A</span> <span m="1991990">time</span> <span m="1992280">slot,</span> <span m="1992990">OK.</span> <span m="1994080">So</span> <span m="1994560">let's</span> <span m="1994850">call</span> <span m="1995180">this</span> <span m="1995410">time</span> <span m="1995660">slot</span> <span m="1996050">C1,</span> <span m="1997990">C2,</span> <span m="1999170">C3,</span> <span m="2000680">C4,</span> <span m="2001210">C5,</span> <span m="2001850">so</span> <span m="2002530">there's</span> <span m="2002690">five</span> <span m="2003010">possible</span> <span m="2003430">colors.</span> <span m="2003850">Now</span> <span m="2003980">of</span> <span m="2004060">course,</span> <span m="2004420">we</span> <span m="2004480">could</span> <span m="2004730">color</span> <span m="2005020">this</span> <span m="2005230">graph</span> <span m="2005570">with</span> <span m="2005720">five</span> <span m="2006000">colors,</span> <span m="2006800">every</span> <span m="2007070">node</span> <span m="2007340">could</span> <span m="2007500">just</span> <span m="2007700">get</span> <span m="2007820">its</span> <span m="2007960">own</span> <span m="2008160">color.</span> <span m="2009150">But</span> <span m="2009340">then</span> <span m="2009950">somebody's</span> <span m="2010410">taking</span> <span m="2010700">their</span> <span m="2010870">exam</span> <span m="2011140">from</span> <span m="2012180">1:00</span> <span m="2012360">to</span> <span m="2012440">3:00</span> <span m="2012650">AM,</span> <span m="2013060">and</span> <span m="2013990">that's</span> <span m="2014240">a</span> <span m="2014440">bit</span> <span m="2014640">of</span> <span m="2014720">a</span> <span m="2014800">pain.</span> </p>
<p><span m="2016880">Let's</span> <span m="2017190">see</span> <span m="2018550">if</span> <span m="2018710">we</span> <span m="2018790">can</span> <span m="2018880">do</span> <span m="2019010">less</span> <span m="2019240">than</span> <span m="2019390">five.</span> <span m="2021360">Let's</span> <span m="2021550">say</span> <span m="2021660">I</span> <span m="2021730">give</span> <span m="2021990">this</span> <span m="2022260">color</span> <span m="2022600">one,</span> <span m="2024960">let's</span> <span m="2025290">give</span> <span m="2025420">this</span> <span m="2025600">one</span> <span m="2025720">color</span> <span m="2026010">one,</span> <span m="2027140">that's</span> <span m="2027380">OK,</span> <span m="2027740">because</span> <span m="2027890">they're</span> <span m="2027980">not</span> <span m="2028160">connected.</span> <span m="2030630">I</span> <span m="2030740">can't</span> <span m="2031040">give</span> <span m="2031210">this</span> <span m="2031400">one</span> <span m="2031560">color</span> <span m="2031870">one,</span> <span m="2032890">so</span> <span m="2033070">I</span> <span m="2033140">give</span> <span m="2033300">it</span> <span m="2033370">color</span> <span m="2033670">two,</span> <span m="2034040">say.</span> <span m="2035880">Now</span> <span m="2036070">this</span> <span m="2036310">one</span> <span m="2036500">I</span> <span m="2036560">can't</span> <span m="2036910">give</span> <span m="2036990">color</span> <span m="2037280">one,</span> <span m="2037670">because</span> <span m="2037850">this</span> <span m="2038020">guy</span> <span m="2038170">got</span> <span m="2038450">it,</span> <span m="2038810">he</span> <span m="2039060">can't</span> <span m="2039150">get</span> <span m="2039420">color</span> <span m="2039690">two,</span> <span m="2039820">because</span> <span m="2040070">that</span> <span m="2040120">guy</span> <span m="2040400">got</span> <span m="2040480">it.</span> <span m="2040570">So</span> <span m="2040630">it</span> <span m="2040800">give</span> <span m="2040860">it</span> <span m="2041140">color</span> <span m="2041495">three.</span> <span m="2043020">And</span> <span m="2044570">well,</span> <span m="2045370">I</span> <span m="2045470">can't</span> <span m="2045810">do</span> <span m="2046430">one,</span> <span m="2046810">two,</span> <span m="2047240">or</span> <span m="2047530">three</span> <span m="2047870">here,</span> <span m="2048159">so</span> <span m="2048270">I</span> <span m="2048320">gotta</span> <span m="2048530">go</span> <span m="2048650">to</span> <span m="2048719">color</span> <span m="2049000">four.</span> </p>
<p><span m="2051020">All</span> <span m="2051170">right</span> <span m="2051320">so</span> <span m="2051420">6042</span> <span m="2051765">will</span> <span m="2052670">get</span> <span m="2053719">the</span> <span m="2053840">11:00</span> <span m="2054600">PM</span> <span m="2055000">to</span> <span m="2055110">1:00</span> <span m="2055280">AM</span> <span m="2055570">slot,</span> <span m="2057800">not</span> <span m="2058120">so</span> <span m="2058300">good.</span> <span m="2059443">Can</span> <span m="2060210">we</span> <span m="2060340">do</span> <span m="2060449">any</span> <span m="2060590">better?</span> <span m="2061916">Can</span> <span m="2062370">we</span> <span m="2062469">get</span> <span m="2062620">away</span> <span m="2062929">with</span> <span m="2063219">three</span> <span m="2063550">colors.</span> <span m="2066139">Some</span> <span m="2066460">say</span> <span m="2066710">yes,</span> <span m="2067090">some</span> <span m="2067179">say</span> <span m="2067409">no.</span> <span m="2069870">How</span> <span m="2070000">many</span> <span m="2070159">people</span> <span m="2070449">think</span> <span m="2070690">you</span> <span m="2070810">can</span> <span m="2071020">do</span> <span m="2072150">three</span> <span m="2072500">colors</span> <span m="2073030">on</span> <span m="2073170">this</span> <span m="2073340">graph?</span> <span m="2073690">A</span> <span m="2074130">bunch.</span> <span m="2074830">How</span> <span m="2075010">many</span> <span m="2075179">think</span> <span m="2075360">you</span> <span m="2075449">can't</span> <span m="2075710">do</span> <span m="2075830">any</span> <span m="2075940">better?</span> <span m="2077402">All</span> <span m="2077840">right,</span> <span m="2078090">the</span> <span m="2078210">vote</span> <span m="2078670">is</span> <span m="2079139">mostly</span> <span m="2079280">for</span> <span m="2079760">three.</span> <span m="2080270">Let's</span> <span m="2080570">see.</span> <span m="2082060">Any</span> <span m="2082210">ideas?</span> <span m="2082639">Anybody</span> <span m="2083060">see</span> <span m="2083250">how</span> <span m="2083380">to</span> <span m="2083480">do</span> <span m="2083610">three?</span> <span m="2087020">Yeah?</span> </p>
<p><span m="2087275">AUDIENCE: Assign</span> <span m="2087530">C4</span> <span m="2087974">to</span> <span m="2088418">6034 .</span> </p>
<p><span m="2090199">PROFESSOR: Assign</span> <span m="2090590">C4</span> <span m="2091550">to</span> <span m="2091670">6043.</span> </p>
<p><span m="2092550">AUDIENCE: Or</span> <span m="2092925">C1</span> <span m="2093300">to</span> <span m="2093585">6042.</span> </p>
<p><span m="2095889">PROFESSOR: C--</span> <span m="2096570">I</span> <span m="2096690">can't</span> <span m="2096989">do</span> <span m="2097110">see</span> <span m="2097350">C1</span> <span m="2097570">to</span> <span m="2097640">6042.</span> <span m="2099580">It</span> <span m="2099710">crashes,</span> <span m="2099975">but</span> <span m="2100240">can</span> <span m="2100560">I</span> <span m="2100630">do--</span> <span m="2101100">yeah?</span> <span m="2101610">Put</span> </p>
<p><span m="2102060">AUDIENCE: C1</span> <span m="2102510">in</span> <span m="2102865">6003.</span> </p>
<p><span m="2104600">PROFESSOR: C1</span> <span m="2104970">in</span> <span m="2105230">6003.</span> </p>
<p><span m="2105350">AUDIENCE: And</span> <span m="2105765">get</span> <span m="2106180">rid</span> <span m="2106595">of</span> <span m="2107010">C1</span> <span m="2107491">in</span> <span m="2107972">6034.</span> </p>
<p><span m="2110380">PROFESSOR: Get</span> <span m="2110570">rid</span> <span m="2110800">of--</span> </p>
<p><span m="2111180">AUDIENCE: Make</span> <span m="2111560">it</span> <span m="2111900">C2.</span> </p>
<p><span m="2112580">PROFESSOR: Make</span> <span m="2112790">this</span> <span m="2112960">a</span> <span m="2113435">C2.</span> <span m="2115070">Oh,</span> <span m="2116540">yeah.</span> <span m="2117362">All</span> <span m="2117834">right,</span> <span m="2118780">these</span> <span m="2119160">got</span> <span m="2119420">C1,</span> <span m="2119920">they're</span> <span m="2120060">not</span> <span m="2120250">adjacent.</span> <span m="2121520">These</span> <span m="2121770">got</span> <span m="2122010">C2,</span> <span m="2122510">they're</span> <span m="2122720">not</span> <span m="2122910">adjacent.</span> <span m="2123740">This</span> <span m="2124140">can</span> <span m="2124330">now</span> <span m="2124640">get</span> <span m="2124800">C3.</span> <span m="2126380">So</span> <span m="2126630">we</span> <span m="2126740">can</span> <span m="2126850">have</span> <span m="2127030">our</span> <span m="2127150">exam</span> <span m="2127560">from</span> <span m="2127690">9:00</span> <span m="2127950">to</span> <span m="2128050">11:00,</span> <span m="2128660">which</span> <span m="2128850">is</span> <span m="2128960">better.</span> <span m="2129990">All</span> <span m="2130380">right,</span> <span m="2131030">can</span> <span m="2131170">anybody</span> <span m="2131520">do</span> <span m="2131690">it</span> <span m="2131740">in</span> <span m="2131840">two</span> <span m="2132060">colors?</span> <span m="2140780">Can</span> <span m="2140940">anybody</span> <span m="2141960">offer</span> <span m="2142290">a</span> <span m="2143000">reason</span> <span m="2143350">why</span> <span m="2143590">two</span> <span m="2143800">colors</span> <span m="2144170">may</span> <span m="2144290">not</span> <span m="2144480">be</span> <span m="2144610">possible?</span> <span m="2147700">Yeah?</span> </p>
<p><span m="2148035">AUDIENCE: Because</span> <span m="2148370">let's</span> <span m="2148824">say</span> <span m="2149278">you</span> <span m="2149732">could</span> <span m="2150186">do</span> <span m="2150640">it</span> <span m="2151094">with</span> <span m="2151550">two</span> <span m="2151650">colors.</span> </p>
<p><span m="2151840">PROFESSOR: Yep.</span> </p>
<p><span m="2152200">AUDIENCE: 6041</span> <span m="2153140">and</span> <span m="2154568">6002</span> <span m="2156062">have</span> <span m="2156560">to</span> <span m="2157058">be</span> <span m="2157560">different</span> <span m="2157865">colors.</span> </p>
<p><span m="2158170">PROFESSOR: Yes.</span> </p>
<p><span m="2159830">AUDIENCE: 6042</span> <span m="2160150">can't</span> <span m="2160550">be</span> <span m="2160950">C1,</span> <span m="2161170">and</span> <span m="2161450">it</span> <span m="2161860">can't</span> <span m="2162270">be</span> <span m="2162680">C2.</span> </p>
<p><span m="2163500">PROFESSOR: Yeah,</span> <span m="2163875">good.</span> <span m="2165300">So</span> <span m="2165380">you</span> <span m="2165520">can't</span> <span m="2165850">in</span> <span m="2165950">two</span> <span m="2166130">colors,</span> <span m="2166540">because</span> <span m="2167080">these</span> <span m="2167420">three</span> <span m="2167680">guys</span> <span m="2169130">would</span> <span m="2169280">violate</span> <span m="2169890">that.</span> <span m="2170670">You've</span> <span m="2170760">got</span> <span m="2170890">a</span> <span m="2170940">triangle</span> <span m="2171520">here.</span> <span m="2171990">Each</span> <span m="2172240">one</span> <span m="2172440">of</span> <span m="2172500">these</span> <span m="2172690">guys</span> <span m="2173000">has</span> <span m="2173210">to</span> <span m="2173290">be</span> <span m="2173400">different</span> <span m="2173810">than</span> <span m="2173920">the</span> <span m="2174030">other</span> <span m="2174210">two.</span> <span m="2175550">So</span> <span m="2175650">two</span> <span m="2175830">colors</span> <span m="2176180">can't</span> <span m="2176430">work.</span> <span m="2176920">You've</span> <span m="2177000">got</span> <span m="2177130">to</span> <span m="2177340">have</span> <span m="2177600">at</span> <span m="2177860">least</span> <span m="2178120">three</span> <span m="2178500">in</span> <span m="2178620">this</span> <span m="2178790">case.</span> <span m="2179740">So</span> <span m="2179940">three</span> <span m="2180250">is</span> <span m="2180390">optimal.</span> <span m="2180760">We</span> <span m="2181130">have</span> <span m="2181350">just</span> <span m="2181580">shown</span> <span m="2181900">for</span> <span m="2182120">this</span> <span m="2182400">graph,</span> <span m="2184000">the</span> <span m="2184110">chromatic</span> <span m="2184570">number</span> <span m="2185020">is</span> <span m="2185390">three.</span> </p>
<p><span m="2186836">All</span> <span m="2187320">right,</span> <span m="2187470">now</span> <span m="2188070">in</span> <span m="2188260">general</span> <span m="2189610">doing</span> <span m="2189860">what</span> <span m="2190000">we</span> <span m="2190140">just</span> <span m="2190420">did</span> <span m="2190830">is</span> <span m="2191040">very</span> <span m="2191330">hard.</span> <span m="2193300">No</span> <span m="2193640">one</span> <span m="2193840">knows</span> <span m="2194240">a</span> <span m="2194300">fast</span> <span m="2194730">algorithm</span> <span m="2195220">for</span> <span m="2195340">determining</span> <span m="2195680">the</span> <span m="2196020">chromatic</span> <span m="2196490">number.</span> <span m="2198560">In</span> <span m="2198590">fact,</span> <span m="2199360">it's</span> <span m="2199420">a</span> <span m="2199470">weird</span> <span m="2199760">kind</span> <span m="2199920">of</span> <span m="2200010">problem,</span> <span m="2200370">because</span> <span m="2201590">it's</span> <span m="2201820">easy</span> <span m="2202200">enough</span> <span m="2202390">to</span> <span m="2202530">check</span> <span m="2203000">that</span> <span m="2203110">a</span> <span m="2203200">coloring</span> <span m="2203700">is</span> <span m="2203830">OK.</span> <span m="2204650">If</span> <span m="2204760">somebody</span> <span m="2205030">put</span> <span m="2205190">a</span> <span m="2205240">coloring</span> <span m="2205550">on</span> <span m="2205630">the</span> <span m="2205710">board,</span> <span m="2206070">you</span> <span m="2206140">can</span> <span m="2206270">check,</span> <span m="2206530">oh</span> <span m="2206750">that</span> <span m="2206950">works</span> <span m="2207670">really</span> <span m="2207930">simply.</span> <span m="2208420">Just</span> <span m="2208610">check</span> <span m="2208880">every</span> <span m="2209140">edge,</span> <span m="2209820">and</span> <span m="2209900">make</span> <span m="2210050">sure</span> <span m="2210170">the</span> <span m="2210280">colors</span> <span m="2210640">are</span> <span m="2210690">different.</span> </p>
<p><span m="2212070">But</span> <span m="2212250">figuring</span> <span m="2212620">it</span> <span m="2212700">out,</span> <span m="2213760">as</span> <span m="2213930">best</span> <span m="2214200">we</span> <span m="2214300">know,</span> <span m="2214590">you've</span> <span m="2214710">got</span> <span m="2214820">to</span> <span m="2214890">try</span> <span m="2215420">an</span> <span m="2215510">exponential</span> <span m="2216100">number</span> <span m="2216320">of</span> <span m="2216370">possibilities.</span> <span m="2217840">So</span> <span m="2217880">if</span> <span m="2217960">I</span> <span m="2218090">had</span> <span m="2218160">100</span> <span m="2218740">nodes</span> <span m="2219090">here,</span> <span m="2220070">my</span> <span m="2220540">running</span> <span m="2221030">time</span> <span m="2221280">of</span> <span m="2221360">the</span> <span m="2221520">algorithm</span> <span m="2221840">to</span> <span m="2221920">check</span> <span m="2222270">all</span> <span m="2222510">the</span> <span m="2222600">possibilities</span> <span m="2222920">would</span> <span m="2223240">be</span> <span m="2223480">exponential</span> <span m="2223810">and</span> <span m="2224140">a</span> <span m="2224395">hundred.</span> <span m="2225050">Yeah?</span> </p>
<p><span m="2225300">AUDIENCE: Can</span> <span m="2225796">that</span> <span m="2226292">number</span> <span m="2226788">just</span> <span m="2227284">like</span> <span m="2227780">the</span> <span m="2228276">highest</span> <span m="2228772">degree</span> <span m="2229268">of</span> <span m="2229764">each</span> <span m="2230260">node,</span> <span m="2230756">or</span> <span m="2231252">nodes.</span> </p>
<p><span m="2231750">PROFESSOR: Uh</span> <span m="2232680">no.</span> <span m="2233015">But</span> <span m="2233350">it's</span> <span m="2233590">no</span> <span m="2234220">worse</span> <span m="2234680">than</span> <span m="2235090">something</span> <span m="2235380">like</span> <span m="2235580">that,</span> <span m="2235770">as</span> <span m="2235870">we'll</span> <span m="2236030">see</span> <span m="2236300">a</span> <span m="2236370">few</span> <span m="2236530">minutes.</span> <span m="2237130">That's</span> <span m="2237330">a</span> <span m="2237390">great</span> <span m="2237820">observation.</span> <span m="2238750">And</span> <span m="2238870">we're</span> <span m="2238960">going</span> <span m="2239040">to</span> <span m="2239100">come</span> <span m="2239270">back</span> <span m="2239510">to</span> <span m="2239570">that</span> <span m="2240250">in</span> <span m="2240350">a</span> <span m="2240420">few</span> <span m="2240590">minutes.</span> <span m="2241210">But</span> <span m="2241330">it's</span> <span m="2241510">not</span> <span m="2241890">just</span> <span m="2242190">that.</span> </p>
<p><span m="2244180">OK</span> <span m="2245180">now</span> <span m="2246270">in</span> <span m="2246390">fact</span> <span m="2247960">even</span> <span m="2248330">figuring</span> <span m="2248810">out</span> <span m="2249040">for</span> <span m="2249230">an</span> <span m="2249310">arbitrary</span> <span m="2249750">graph</span> <span m="2250140">if</span> <span m="2250290">three</span> <span m="2250800">colors</span> <span m="2251720">can</span> <span m="2251870">be</span> <span m="2251990">done,</span> <span m="2253080">called</span> <span m="2253320">the</span> <span m="2253400">three-coloring</span> <span m="2254050">problem,</span> <span m="2255020">that's</span> <span m="2255300">really</span> <span m="2255550">hard.</span> <span m="2256290">No</span> <span m="2256490">one</span> <span m="2256620">knows</span> <span m="2256810">how</span> <span m="2256930">to</span> <span m="2257010">solve</span> <span m="2257300">that</span> <span m="2257570">in</span> <span m="2257670">less</span> <span m="2257900">than</span> <span m="2258020">exponential</span> <span m="2258580">time.</span> <span m="2259570">In</span> <span m="2259720">fact,</span> <span m="2261080">one</span> <span m="2261210">of</span> <span m="2261260">these</span> <span m="2261630">NP-complete</span> <span m="2262420">problems</span> <span m="2263000">is</span> <span m="2263130">what</span> <span m="2263290">it's</span> <span m="2263410">called.</span> <span m="2264340">How</span> <span m="2264480">many</span> <span m="2264690">people</span> <span m="2265120">here</span> <span m="2265370">don't</span> <span m="2265770">know</span> <span m="2265890">about</span> <span m="2266150">NP-completeness?</span> <span m="2267870">Is</span> <span m="2267980">everybody--</span> <span m="2268260">all</span> <span m="2268600">right</span> <span m="2268940">so</span> <span m="2269280">all</span> <span m="2269540">of</span> <span m="2270040">you</span> <span m="2270230">haven't</span> <span m="2270380">seen</span> <span m="2270705">NP-completeness.</span> </p>
<p><span m="2271030">OK</span> <span m="2271320">so</span> <span m="2272180">there</span> <span m="2272520">is</span> <span m="2272700">a</span> <span m="2273620">class</span> <span m="2274270">of</span> <span m="2274600">thousands</span> <span m="2275280">of</span> <span m="2275370">problems--</span> <span m="2276120">in</span> <span m="2276190">fact</span> <span m="2276380">there's</span> <span m="2276550">books</span> <span m="2277250">list</span> <span m="2278040">these</span> <span m="2278250">1,000</span> <span m="2278670">problems--</span> <span m="2279250">that</span> <span m="2279660">are</span> <span m="2279930">all</span> <span m="2280230">NP-complete,</span> <span m="2281230">somebody's</span> <span m="2281570">proved</span> <span m="2281950">they</span> <span m="2282050">belong</span> <span m="2282340">in</span> <span m="2282400">the</span> <span m="2282470">class.</span> <span m="2283000">And</span> <span m="2283100">what</span> <span m="2283150">that</span> <span m="2283350">means</span> <span m="2284610">is</span> <span m="2284860">that</span> <span m="2285480">if</span> <span m="2285600">somebody</span> <span m="2285960">gave</span> <span m="2286290">you</span> <span m="2286440">a</span> <span m="2286500">solution,</span> <span m="2287970">like</span> <span m="2288150">a</span> <span m="2288220">coloring</span> <span m="2288720">here,</span> <span m="2289300">it's</span> <span m="2289510">easy</span> <span m="2289840">to</span> <span m="2289940">check</span> <span m="2290380">really</span> <span m="2290670">quickly</span> <span m="2291420">if</span> <span m="2291560">it's</span> <span m="2291710">valid.</span> <span m="2293850">But</span> <span m="2294210">figuring</span> <span m="2294640">it</span> <span m="2294720">out</span> <span m="2294900">is</span> <span m="2295010">really</span> <span m="2295280">hard.</span> <span m="2296040">And</span> <span m="2296080">if</span> <span m="2296190">you</span> <span m="2296360">figured</span> <span m="2296800">out</span> <span m="2296980">how</span> <span m="2297160">to</span> <span m="2297250">solve</span> <span m="2297790">one</span> <span m="2298120">of</span> <span m="2298180">those</span> <span m="2298420">thousands</span> <span m="2298850">of</span> <span m="2298930">problems,</span> <span m="2299870">like</span> <span m="2300050">suddenly</span> <span m="2300390">you</span> <span m="2300500">figured</span> <span m="2300800">out</span> <span m="2300960">how</span> <span m="2301120">to</span> <span m="2301230">tell</span> <span m="2301450">if</span> <span m="2301570">any</span> <span m="2301770">graph</span> <span m="2302120">could</span> <span m="2302260">work</span> <span m="2302470">with</span> <span m="2302590">three</span> <span m="2302740">colors,</span> <span m="2303920">you</span> <span m="2304070">would</span> <span m="2304220">solve</span> <span m="2304760">automatically</span> <span m="2305610">all</span> <span m="2305940">other</span> <span m="2306190">thousands</span> <span m="2306740">in</span> <span m="2306830">the</span> <span m="2306910">book.</span> </p>
<p><span m="2308510">So</span> <span m="2308660">it's</span> <span m="2308750">this</span> <span m="2308910">book</span> <span m="2309190">of</span> <span m="2309270">problems</span> <span m="2309670">you</span> <span m="2309790">will</span> <span m="2309900">constantly</span> <span m="2310470">run</span> <span m="2310650">into</span> <span m="2310980">in</span> <span m="2311110">your</span> <span m="2311510">career</span> <span m="2311910">in</span> <span m="2312020">computer</span> <span m="2312370">science.</span> <span m="2314060">And</span> <span m="2314180">it's</span> <span m="2314320">bad</span> <span m="2314630">when</span> <span m="2314730">you</span> <span m="2314830">run</span> <span m="2314990">into</span> <span m="2315180">one,</span> <span m="2315410">because</span> <span m="2316150">there's</span> <span m="2316350">no</span> <span m="2316580">good</span> <span m="2316780">algorithm</span> <span m="2317090">to</span> <span m="2317140">solve</span> <span m="2317470">it</span> <span m="2317580">known.</span> <span m="2318200">But</span> <span m="2318380">if</span> <span m="2318440">you</span> <span m="2318560">just</span> <span m="2318830">solved</span> <span m="2319160">one</span> <span m="2319410">of</span> <span m="2319480">them,</span> <span m="2320065">the</span> <span m="2320360">other</span> <span m="2320640">thousands</span> <span m="2321140">would</span> <span m="2321300">suddenly</span> <span m="2321720">be</span> <span m="2321830">solvable</span> <span m="2322340">quickly.</span> <span m="2323620">Even</span> <span m="2323930">better,</span> <span m="2324650">you</span> <span m="2324860">win</span> <span m="2325040">a</span> <span m="2325090">million</span> <span m="2325470">dollar</span> <span m="2325770">prize.</span> <span m="2326540">One</span> <span m="2326680">of</span> <span m="2326730">these</span> <span m="2327060">Millennium</span> <span m="2327500">Prizes</span> <span m="2327880">we</span> <span m="2328000">talked</span> <span m="2328260">about</span> <span m="2328500">the</span> <span m="2328590">first</span> <span m="2328880">lecture.</span> </p>
<p><span m="2330860">Even</span> <span m="2331090">if</span> <span m="2331180">you</span> <span m="2331280">show</span> <span m="2332100">you</span> <span m="2332230">can't</span> <span m="2332620">find</span> <span m="2332850">a</span> <span m="2332900">fast</span> <span m="2333240">algorithm</span> <span m="2333550">for</span> <span m="2333660">one</span> <span m="2333880">of</span> <span m="2333950">them,</span> <span m="2334500">that</span> <span m="2334700">means</span> <span m="2334980">that</span> <span m="2335130">known</span> <span m="2335450">of</span> <span m="2335540">them</span> <span m="2335680">have</span> <span m="2335910">fast</span> <span m="2336250">algorithms,</span> <span m="2337000">and</span> <span m="2337230">you</span> <span m="2337320">also</span> <span m="2337570">get</span> <span m="2337680">a</span> <span m="2337720">million</span> <span m="2337970">dollars.</span> <span m="2339430">So</span> <span m="2339540">this</span> <span m="2339710">is</span> <span m="2339820">the</span> <span m="2339940">central</span> <span m="2340500">problem</span> <span m="2340810">in</span> <span m="2340890">computer</span> <span m="2341210">science,</span> <span m="2341540">and</span> <span m="2341650">theory</span> <span m="2341980">computing,</span> <span m="2342810">is</span> <span m="2343210">whether</span> <span m="2343530">or</span> <span m="2343570">not</span> <span m="2343810">you</span> <span m="2343930">could</span> <span m="2344070">solve</span> <span m="2344410">these</span> <span m="2344590">NP-complete</span> <span m="2345110">problems.</span> </p>
<p><span m="2347060">Now</span> <span m="2347220">actually</span> <span m="2347590">lots</span> <span m="2347920">of</span> <span m="2348010">people</span> <span m="2348310">have</span> <span m="2348470">claim</span> <span m="2348890">to</span> <span m="2348980">do</span> <span m="2349190">it.</span> <span m="2349770">And</span> <span m="2349850">in</span> <span m="2349900">fact,</span> <span m="2350120">there</span> <span m="2350190">was</span> <span m="2350300">a</span> <span m="2350360">lot</span> <span m="2350650">of</span> <span m="2350750">buzz</span> <span m="2351570">in</span> <span m="2351670">the</span> <span m="2351760">community</span> <span m="2352300">about</span> <span m="2352510">a</span> <span m="2352540">month</span> <span m="2352870">ago</span> <span m="2353610">when</span> <span m="2353800">actually</span> <span m="2354160">a</span> <span m="2354300">reputable</span> <span m="2354790">researcher</span> <span m="2355110">at</span> <span m="2355430">HP</span> <span m="2355800">Labs</span> <span m="2356145">said</span> <span m="2356490">he'd</span> <span m="2356820">done</span> <span m="2357140">it.</span> <span m="2357370">He</span> <span m="2357770">proved</span> <span m="2359140">that</span> <span m="2359550">you</span> <span m="2359670">can't</span> <span m="2360020">solve</span> <span m="2360380">NP-complete</span> <span m="2360930">problems.</span> <span m="2361630">And</span> <span m="2361810">he</span> <span m="2361880">got</span> <span m="2362090">people</span> <span m="2362350">going</span> <span m="2362560">for</span> <span m="2362710">probably</span> <span m="2362870">at</span> <span m="2363340">least</span> <span m="2363600">a</span> <span m="2363670">week,</span> <span m="2364100">until</span> <span m="2364710">they</span> <span m="2364800">discovered</span> <span m="2365180">a</span> <span m="2365230">fatal</span> <span m="2365510">flaw.</span> <span m="2366420">And</span> <span m="2366780">the</span> <span m="2367070">proof</span> <span m="2367320">was</span> <span m="2367640">actually</span> <span m="2367940">bogus.</span> <span m="2369550">So</span> <span m="2369900">no</span> <span m="2370080">one</span> <span m="2370230">still</span> <span m="2370500">knows</span> <span m="2371020">if</span> <span m="2371170">you</span> <span m="2371280">can</span> <span m="2371450">solve</span> <span m="2371770">these</span> <span m="2371950">NP-complete</span> <span m="2372530">problems</span> <span m="2372960">quickly.</span> </p>
<p><span m="2374770">Now</span> <span m="2375730">the</span> <span m="2375860">problem</span> <span m="2376270">is,</span> <span m="2376520">in</span> <span m="2376620">practice,</span> <span m="2377630">you</span> <span m="2377790">run</span> <span m="2377980">into</span> <span m="2378170">these</span> <span m="2378380">things</span> <span m="2378620">all</span> <span m="2378750">the</span> <span m="2378850">time,</span> <span m="2379160">like</span> <span m="2379400">MIT</span> <span m="2379890">really</span> <span m="2380130">does</span> <span m="2380430">have</span> <span m="2380580">to</span> <span m="2380670">schedule</span> <span m="2381040">the</span> <span m="2381150">exams.</span> <span m="2382520">So</span> <span m="2382560">you've</span> <span m="2382630">got</span> <span m="2382780">to</span> <span m="2382840">do</span> <span m="2382960">something.</span> <span m="2383680">You</span> <span m="2383750">can't</span> <span m="2383990">just</span> <span m="2384200">go</span> <span m="2384330">say,</span> <span m="2384610">hey</span> <span m="2384750">it's</span> <span m="2384900">NP-complete,</span> <span m="2385530">so</span> <span m="2386160">no</span> <span m="2386320">exams</span> <span m="2386690">this</span> <span m="2386850">year,</span> <span m="2387210">or</span> <span m="2388510">whatever.</span> <span m="2388870">That's</span> <span m="2389370">not</span> <span m="2389540">going</span> <span m="2389640">to</span> <span m="2389700">fly,</span> <span m="2390080">so</span> <span m="2390230">you</span> <span m="2390840">got</span> <span m="2390980">to</span> <span m="2391040">do</span> <span m="2391160">something.</span> </p>
<p><span m="2392970">So</span> <span m="2393250">now</span> <span m="2394060">this</span> <span m="2394250">is</span> <span m="2394360">a</span> <span m="2394420">problem--</span> <span m="2394860">many</span> <span m="2395120">of</span> <span m="2395170">you</span> <span m="2395320">when</span> <span m="2395410">you</span> <span m="2395500">go</span> <span m="2395680">into</span> <span m="2395850">careers,</span> <span m="2396360">you're</span> <span m="2396430">going</span> <span m="2396530">to</span> <span m="2396690">be</span> <span m="2396840">faced</span> <span m="2397260">with</span> <span m="2397380">this.</span> <span m="2398510">You</span> <span m="2398610">got</span> <span m="2398710">to</span> <span m="2398820">do</span> <span m="2398920">something.</span> <span m="2400250">Any</span> <span m="2400490">thoughts</span> <span m="2400950">about</span> <span m="2401540">an</span> <span m="2401690">algorithm</span> <span m="2403450">for</span> <span m="2404710">coloring</span> <span m="2405180">graphs</span> <span m="2405650">that</span> <span m="2405780">might</span> <span m="2406010">use</span> <span m="2406190">a</span> <span m="2406250">small</span> <span m="2406610">number</span> <span m="2406830">of</span> <span m="2406900">colors?</span> <span m="2407880">It</span> <span m="2408740">doesn't</span> <span m="2408930">have</span> <span m="2409120">to</span> <span m="2409210">always</span> <span m="2409570">work,</span> <span m="2409950">or</span> <span m="2410430">you're</span> <span m="2410490">going</span> <span m="2410610">to</span> <span m="2410730">win</span> <span m="2410770">a</span> <span m="2410920">lot</span> <span m="2410980">of</span> <span m="2411180">money</span> <span m="2411270">if</span> <span m="2411360">it</span> <span m="2411690">does.</span> <span m="2411960">But</span> <span m="2412220">a</span> <span m="2412320">simple</span> <span m="2412800">algorithm,</span> <span m="2413890">you</span> <span m="2413990">can't</span> <span m="2414300">take</span> <span m="2414800">either</span> <span m="2415000">the</span> <span m="2415110">100</span> <span m="2415410">steps.</span> <span m="2416590">You</span> <span m="2416670">got</span> <span m="2416790">to</span> <span m="2416830">be</span> <span m="2416930">linear,</span> <span m="2417400">probably,</span> <span m="2417870">or</span> <span m="2417960">quadratic</span> <span m="2418540">time.</span> <span m="2419690">That</span> <span m="2419810">could</span> <span m="2420040">get</span> <span m="2420230">you</span> <span m="2420360">a</span> <span m="2420420">small</span> <span m="2420720">number</span> <span m="2420920">of</span> <span m="2421000">colors.</span> <span m="2422800">Any</span> <span m="2422940">thoughts</span> <span m="2424520">about</span> <span m="2424740">what</span> <span m="2424880">you'd</span> <span m="2424980">do?</span> <span m="2426100">Yeah?</span> </p>
<p><span m="2426360">AUDIENCE: The</span> <span m="2426620">number</span> <span m="2427050">of</span> <span m="2427480">degrees</span> <span m="2427910">and</span> <span m="2428340">nodes?</span> </p>
<p><span m="2428770">PROFESSOR: The</span> <span m="2428890">number--</span> <span m="2429660">what</span> <span m="2429900">about</span> <span m="2430190">it?</span> </p>
<p><span m="2430590">AUDIENCE: The</span> <span m="2431028">highest</span> <span m="2431466">degree</span> <span m="2431904">and</span> <span m="2432342">that</span> <span m="2432780">node,</span> <span m="2433220">the</span> <span m="2433250">6042</span> <span m="2433725">is</span> <span m="2434200">[INAUDIBLE].</span> </p>
<p><span m="2435150">PROFESSOR: Yeah.</span> </p>
<p><span m="2435465">AUDIENCE: So</span> <span m="2435780">you</span> <span m="2436266">could</span> <span m="2436752">use</span> <span m="2437238">that.</span> </p>
<p><span m="2438700">PROFESSOR: Good,</span> <span m="2439510">all</span> <span m="2439650">right.</span> <span m="2440210">So</span> <span m="2440320">what</span> <span m="2440440">do</span> <span m="2440540">I</span> <span m="2440620">do</span> <span m="2440880">with</span> <span m="2441340">that--</span> <span m="2441520">so</span> <span m="2441730">I</span> <span m="2441840">found</span> <span m="2442200">a</span> <span m="2442830">node</span> <span m="2443110">with</span> <span m="2443220">a</span> <span m="2443320">high</span> <span m="2443440">degree,</span> <span m="2444150">there's</span> <span m="2445130">three</span> <span m="2445360">of</span> <span m="2445420">them</span> <span m="2445560">have</span> <span m="2445730">degree</span> <span m="2446050">three</span> <span m="2446330">here.</span> <span m="2447030">What</span> <span m="2447200">do</span> <span m="2447290">I</span> <span m="2447380">do</span> <span m="2447600">with</span> <span m="2447720">them?</span> </p>
<p><span m="2448806">AUDIENCE: Pick</span> <span m="2449249">a</span> <span m="2449692">different</span> <span m="2450135">color</span> <span m="2450578">to.</span> </p>
<p><span m="2452850">PROFESSOR: Pick</span> <span m="2453020">a</span> <span m="2453080">different</span> <span m="2453410">color,</span> <span m="2453740">that</span> <span m="2453960">means</span> <span m="2454290">I've</span> <span m="2454590">colored</span> <span m="2454930">some</span> <span m="2455100">of</span> <span m="2455150">the</span> <span m="2455290">others.</span> <span m="2455890">If</span> <span m="2456346">I</span> <span m="2456802">pick</span> <span m="2457260">a</span> <span m="2457390">different</span> <span m="2457590">color,</span> <span m="2457620">do</span> <span m="2457970">I</span> <span m="2458280">start</span> <span m="2458750">with</span> <span m="2458940">them,</span> <span m="2459040">or</span> <span m="2459110">do</span> <span m="2459210">I</span> <span m="2459610">finish</span> <span m="2459700">with</span> <span m="2459770">a</span> <span m="2460060">high</span> <span m="2460555">degree</span> <span m="2461050">nodes?</span> <span m="2463030">Because</span> <span m="2463525">you've</span> <span m="2464020">got</span> <span m="2464160">to</span> <span m="2464310">assign</span> <span m="2464680">the</span> <span m="2464770">colors</span> <span m="2465120">to</span> <span m="2465440">them.</span> <span m="2466120">And</span> <span m="2466240">high</span> <span m="2466500">degree</span> <span m="2466760">is</span> <span m="2466880">important</span> <span m="2467230">to</span> <span m="2467270">be</span> <span m="2467360">thinking</span> <span m="2467640">about.</span> <span m="2467890">We're</span> <span m="2468100">going</span> <span m="2468210">to</span> <span m="2468540">prove</span> <span m="2468720">a</span> <span m="2468770">theorem</span> <span m="2469200">in</span> <span m="2469400">just</span> <span m="2469440">a</span> <span m="2469670">minute</span> <span m="2469955">about</span> <span m="2470240">related</span> <span m="2470300">to</span> <span m="2470630">degree</span> <span m="2471060">and</span> <span m="2471395">coloring.</span> </p>
<p><span m="2471730">AUDIENCE: Start</span> <span m="2472168">with</span> <span m="2472606">them.</span> </p>
<p><span m="2474800">PROFESSOR: Start</span> <span m="2475100">with</span> <span m="2475170">them,</span> <span m="2475970">and</span> <span m="2476225">do</span> <span m="2476480">what</span> <span m="2476580">with</span> <span m="2476880">it?</span> <span m="2477310">Color?</span> </p>
<p><span m="2478498">AUDIENCE: Yeah,</span> <span m="2478986">and</span> <span m="2479474">then</span> <span m="2479962">assign</span> <span m="2480450">the</span> <span m="2480938">ones</span> <span m="2481426">that</span> <span m="2481914">aren't</span> <span m="2482402">connected</span> <span m="2482890">[INAUDIBLE]</span> <span m="2483866">to</span> <span m="2484354">the</span> <span m="2484842">same</span> <span m="2485330">slots.</span> </p>
<p><span m="2486310">PROFESSOR: OK,</span> <span m="2486750">so</span> <span m="2487000">I</span> <span m="2487150">could--</span> <span m="2487750">here's</span> <span m="2488145">a</span> <span m="2488540">degree</span> <span m="2488770">of</span> <span m="2488990">theory</span> <span m="2489250">now</span> <span m="2489510">I</span> <span m="2489795">can</span> <span m="2490080">start</span> <span m="2490220">with</span> <span m="2490595">color</span> <span m="2490970">one</span> <span m="2491060">for</span> <span m="2491350">that.</span> <span m="2492630">And</span> <span m="2492860">then</span> <span m="2493350">what</span> <span m="2493480">do</span> <span m="2493560">I</span> <span m="2493630">do</span> <span m="2493760">next?</span> <span m="2494190">I</span> <span m="2494430">pick--</span> <span m="2496040">its</span> <span m="2496300">neighbors</span> <span m="2496620">have</span> <span m="2496750">to</span> <span m="2496840">get</span> <span m="2497230">different</span> <span m="2497580">colors,</span> <span m="2497930">I</span> <span m="2497990">guess.</span> <span m="2498480">You'd</span> <span m="2498610">start</span> <span m="2498850">coloring</span> <span m="2499160">the</span> <span m="2499230">neighbors.</span> </p>
<p><span m="2499830">AUDIENCE: My</span> <span m="2500325">first</span> <span m="2500820">instinct</span> <span m="2501315">would</span> <span m="2501810">be</span> <span m="2502305">to</span> <span m="2502800">color</span> <span m="2503295">all</span> <span m="2503790">the</span> <span m="2504285">[INAUDIBLE].</span> </p>
<p><span m="2508250">PROFESSOR: OK.</span> <span m="2509530">And</span> <span m="2509810">what</span> <span m="2510040">color</span> <span m="2510190">would</span> <span m="2510350">use</span> <span m="2510610">for</span> <span m="2510870">them?</span> </p>
<p><span m="2511320">AUDIENCE: Different</span> <span m="2511805">ones.</span> </p>
<p><span m="2513260">PROFESSOR: Different</span> <span m="2513630">ones</span> <span m="2513920">if</span> <span m="2514010">they're</span> <span m="2514200">connected,</span> <span m="2514660">or</span> <span m="2514940">if</span> <span m="2515220">they're</span> <span m="2515450">not</span> <span m="2515560">connected</span> <span m="2515900">you'd</span> <span m="2516090">still</span> <span m="2516280">use</span> <span m="2516520">different</span> <span m="2516830">ones?</span> </p>
<p><span m="2517790">AUDIENCE: Only</span> <span m="2518270">if</span> <span m="2518750">they're</span> <span m="2519230">connected.</span> </p>
<p><span m="2520670">PROFESSOR: Only</span> <span m="2520950">they're</span> <span m="2521130">connected</span> <span m="2521570">use</span> <span m="2521780">different</span> <span m="2522160">ones.</span> <span m="2522940">And</span> <span m="2523400">so</span> <span m="2523480">if</span> <span m="2523590">they're</span> <span m="2523720">not</span> <span m="2523960">connected,</span> <span m="2524320">you'd</span> <span m="2524370">use</span> <span m="2524690">the</span> <span m="2524790">same</span> <span m="2525230">colors?</span> <span m="2526924">Yeah?</span> <span m="2530190">You're</span> <span m="2530310">going</span> <span m="2530460">close,</span> <span m="2530740">and</span> <span m="2531060">it</span> <span m="2531220">actually</span> <span m="2531480">works</span> <span m="2531720">pretty</span> <span m="2531900">well.</span> <span m="2534620">The</span> <span m="2534800">underlying</span> <span m="2535180">principle</span> <span m="2535660">you're</span> <span m="2535790">sort</span> <span m="2536150">of</span> <span m="2536250">thinking</span> <span m="2536550">about</span> <span m="2536810">here</span> <span m="2537910">is</span> <span m="2538500">you've</span> <span m="2538670">got</span> <span m="2539040">some</span> <span m="2539460">notion</span> <span m="2539820">of</span> <span m="2539900">the</span> <span m="2540420">order</span> <span m="2540850">in</span> <span m="2540910">which</span> <span m="2541090">you're</span> <span m="2541160">going</span> <span m="2541250">to</span> <span m="2541300">process</span> <span m="2541585">your</span> <span m="2541870">graph.</span> <span m="2543410">And</span> <span m="2543530">you're</span> <span m="2543640">going</span> <span m="2543730">to</span> <span m="2543790">start</span> <span m="2544160">with</span> <span m="2544290">a</span> <span m="2544340">high</span> <span m="2544500">degree</span> <span m="2544730">nodes,</span> <span m="2545110">in</span> <span m="2545180">your</span> <span m="2545310">case.</span> <span m="2546530">And</span> <span m="2547210">as</span> <span m="2547450">you</span> <span m="2547550">go</span> <span m="2547660">along,</span> <span m="2548030">you're</span> <span m="2548100">going</span> <span m="2548200">to</span> <span m="2548250">start</span> <span m="2548520">coloring</span> <span m="2548920">the</span> <span m="2549000">nodes.</span> <span m="2550950">And</span> <span m="2551420">you're</span> <span m="2551560">going</span> <span m="2551650">to</span> <span m="2551700">make</span> <span m="2551850">sure</span> <span m="2552090">you</span> <span m="2552230">color</span> <span m="2552510">them</span> <span m="2552660">legally.</span> <span m="2553990">And</span> <span m="2554120">it</span> <span m="2554210">sounds</span> <span m="2554690">like</span> <span m="2554900">you're</span> <span m="2554990">going</span> <span m="2555090">to</span> <span m="2555150">color</span> <span m="2555520">them</span> <span m="2555650">with</span> <span m="2555710">a</span> <span m="2556190">low</span> <span m="2557145">color</span> <span m="2557920">as</span> <span m="2558000">you</span> <span m="2558090">go</span> <span m="2558230">along.</span> </p>
<p><span m="2559270">And</span> <span m="2559460">that</span> <span m="2559670">is</span> <span m="2560000">probably</span> <span m="2560320">the</span> <span m="2560390">most</span> <span m="2560630">basic</span> <span m="2561710">graph</span> <span m="2562050">coloring</span> <span m="2562420">approach.</span> <span m="2564860">And</span> <span m="2565060">almost</span> <span m="2565410">you</span> <span m="2565490">could</span> <span m="2565630">almost</span> <span m="2565840">say</span> <span m="2566080">is</span> <span m="2566150">a</span> <span m="2566460">generic</span> <span m="2566770">approach.</span> <span m="2568560">So</span> <span m="2568620">let's</span> <span m="2568820">define</span> <span m="2569260">that,</span> <span m="2569530">and</span> <span m="2569880">then</span> <span m="2570010">see</span> <span m="2570490">prove</span> <span m="2570920">some</span> <span m="2571120">facts</span> <span m="2571310">about</span> <span m="2571630">it.</span> </p>
<p><span m="2576710">Most</span> <span m="2577250">of</span> <span m="2577320">the</span> <span m="2577390">graph</span> <span m="2577670">coloring</span> <span m="2577980">algorithms</span> <span m="2578340">in</span> <span m="2578430">practice</span> <span m="2578860">are</span> <span m="2578930">based</span> <span m="2579250">on</span> <span m="2579320">this</span> <span m="2579470">approach.</span> <span m="2581070">And</span> <span m="2581210">we're</span> <span m="2581310">going</span> <span m="2581400">to</span> <span m="2581460">call</span> <span m="2581820">it</span> <span m="2581930">the</span> <span m="2582930">basic</span> <span m="2584480">graph</span> <span m="2584790">coloring</span> <span m="2585150">algorithm.</span> <span m="2594240">And</span> <span m="2594975">for</span> <span m="2595340">our</span> <span m="2595650">graph</span> <span m="2596060">G,</span> <span m="2596740">with</span> <span m="2597520">vertices</span> <span m="2597970">V,</span> <span m="2598230">and</span> <span m="2598330">edges</span> <span m="2598400">E.</span> </p>
<p><span m="2600840">So</span> <span m="2600980">the</span> <span m="2601070">first</span> <span m="2601430">step</span> <span m="2602060">is</span> <span m="2602210">going</span> <span m="2602460">to</span> <span m="2602550">be</span> <span m="2602940">to</span> <span m="2603230">order</span> <span m="2605090">the</span> <span m="2605210">nodes</span> <span m="2610240">from</span> <span m="2610400">1</span> <span m="2610650">to</span> <span m="2610730">n.</span> <span m="2614110">Now</span> <span m="2614340">in</span> <span m="2614430">your</span> <span m="2614620">case,</span> <span m="2615050">you</span> <span m="2615280">were</span> <span m="2615565">suggesting</span> <span m="2615850">an</span> <span m="2615920">ordering</span> <span m="2616460">where</span> <span m="2616610">I</span> <span m="2616680">have</span> <span m="2616890">the</span> <span m="2616930">high</span> <span m="2617250">degree</span> <span m="2617580">nodes</span> <span m="2618570">first.</span> <span m="2619700">All</span> <span m="2620170">right.</span> <span m="2621250">But</span> <span m="2621410">for</span> <span m="2621510">now</span> <span m="2621850">we're</span> <span m="2621960">not</span> <span m="2622130">going</span> <span m="2622230">to</span> <span m="2622300">specify</span> <span m="2622860">that.</span> <span m="2623050">We're</span> <span m="2623150">going</span> <span m="2623230">to</span> <span m="2623270">make</span> <span m="2623410">it</span> <span m="2623540">any</span> <span m="2623810">ordering</span> <span m="2625390">you</span> <span m="2625550">want.</span> <span m="2627290">And</span> <span m="2627500">then</span> <span m="2627650">we're</span> <span m="2627730">going</span> <span m="2627820">to</span> <span m="2627860">have</span> <span m="2628220">a</span> <span m="2628330">notion</span> <span m="2628680">of</span> <span m="2628760">an</span> <span m="2628870">order</span> <span m="2630080">on</span> <span m="2630220">the</span> <span m="2630310">colors,</span> <span m="2630890">as</span> <span m="2631070">well.</span> <span m="2638060">And</span> <span m="2638230">I</span> <span m="2638250">don't</span> <span m="2638350">know</span> <span m="2638410">how</span> <span m="2638600">many</span> <span m="2638800">colors,</span> <span m="2639190">but</span> <span m="2639310">they're</span> <span m="2639410">going</span> <span m="2639520">to</span> <span m="2639570">be</span> <span m="2639720">numbered</span> <span m="2640060">1,</span> <span m="2640370">2,</span> <span m="2640630">and</span> <span m="2640830">so</span> <span m="2641010">forth.</span> </p>
<p><span m="2642480">And</span> <span m="2642630">then</span> <span m="2642780">we're</span> <span m="2642870">going</span> <span m="2642960">to</span> <span m="2643020">process</span> <span m="2643640">the</span> <span m="2643730">nodes</span> <span m="2644090">one</span> <span m="2644330">at</span> <span m="2644400">a</span> <span m="2644460">time,</span> <span m="2648773">to</span> <span m="2649740">N.</span> <span m="2651850">We</span> <span m="2652290">color</span> <span m="2652620">the</span> <span m="2652760">nodes,</span> <span m="2653130">what</span> <span m="2653260">is</span> <span m="2653340">step</span> <span m="2653610">I,</span> <span m="2653860">we</span> <span m="2653970">color</span> <span m="2654300">the</span> <span m="2654610">Ith</span> <span m="2655190">node</span> <span m="2655400">V</span> <span m="2655530">sub</span> <span m="2655970">i</span> <span m="2656430">with</span> <span m="2656670">the</span> <span m="2656780">lowest</span> <span m="2657630">legal</span> <span m="2657970">color.</span> <span m="2665240">And</span> <span m="2665380">by</span> <span m="2665500">the</span> <span m="2665570">legal</span> <span m="2665920">I</span> <span m="2665990">mean</span> <span m="2666830">you</span> <span m="2666960">don't</span> <span m="2667340">color</span> <span m="2667680">at</span> <span m="2667750">the</span> <span m="2667830">same</span> <span m="2668130">node</span> <span m="2668490">as</span> <span m="2668810">another</span> <span m="2668970">node</span> <span m="2669220">that's</span> <span m="2669530">already</span> <span m="2669680">been</span> <span m="2669955">colored</span> <span m="2670470">the</span> <span m="2670550">same</span> <span m="2671000">that</span> <span m="2671160">it's</span> <span m="2671640">adjacent</span> <span m="2672121">to.</span> </p>
<p></p>
<p><span m="2680790">All</span> <span m="2680920">right</span> <span m="2681110">so</span> <span m="2681210">let's</span> <span m="2681490">try</span> <span m="2681940">this.</span> <span m="2683290">In</span> <span m="2683370">fact,</span> <span m="2684450">this</span> <span m="2684660">is</span> <span m="2684750">sort</span> <span m="2685050">of</span> <span m="2685090">the</span> <span m="2685250">algorithm</span> <span m="2685880">I</span> <span m="2686030">used</span> <span m="2686720">initially</span> <span m="2687480">to</span> <span m="2687610">color</span> <span m="2688570">exam</span> <span m="2688960">graph</span> <span m="2690400">over</span> <span m="2690620">there.</span> <span m="2694508">All</span> <span m="2694984">right,</span> <span m="2695460">so</span> <span m="2695710">let's</span> <span m="2695930">look</span> <span m="2696130">at</span> <span m="2696200">that.</span> </p>
<p><span m="2698230">So</span> <span m="2698650">let's</span> <span m="2698960">say</span> <span m="2699190">we--</span> <span m="2699580">let</span> <span m="2699640">me</span> <span m="2699750">erase</span> <span m="2700190">the</span> <span m="2700300">colors</span> <span m="2700730">here,</span> <span m="2703780">and</span> <span m="2704150">put</span> <span m="2704310">an</span> <span m="2704410">ordering</span> <span m="2704910">on</span> <span m="2705020">the</span> <span m="2705100">nodes.</span> <span m="2708100">So</span> <span m="2708410">let's</span> <span m="2708660">say</span> <span m="2708790">I</span> <span m="2708900">ordered</span> <span m="2709280">them</span> <span m="2709520">with</span> <span m="2709660">6034</span> <span m="2710930">first,</span> <span m="2712230">so</span> <span m="2712370">this</span> <span m="2712530">would</span> <span m="2712600">be</span> <span m="2712720">V1.</span> <span m="2715060">Then</span> <span m="2715560">6041</span> <span m="2716350">is</span> <span m="2716500">V2.</span> <span m="2718390">Then</span> <span m="2719500">V3,</span> <span m="2721080">V4,</span> <span m="2722920">V5.</span> <span m="2726290">If</span> <span m="2726490">that's</span> <span m="2726750">my</span> <span m="2726920">ordering,</span> <span m="2727950">what</span> <span m="2728170">color</span> <span m="2728480">would</span> <span m="2728620">I</span> <span m="2728730">assign</span> <span m="2729130">to</span> <span m="2729200">6034?</span> </p>
<p><span m="2730552">AUDIENCE: One.</span> </p>
<p><span m="2731730">PROFESSOR: One,</span> <span m="2732120">C1,</span> <span m="2732610">I'd</span> <span m="2732660">color</span> <span m="2733000">it</span> <span m="2733110">first</span> <span m="2733400">to</span> <span m="2733450">get</span> <span m="2733610">C1.</span> <span m="2735140">What</span> <span m="2735340">color</span> <span m="2735660">does</span> <span m="2735780">6041</span> <span m="2736800">get?</span> <span m="2738860">C1,</span> <span m="2739450">as</span> <span m="2739570">well,</span> <span m="2739920">it's</span> <span m="2740040">the</span> <span m="2740130">lowest</span> <span m="2740630">possible</span> <span m="2741050">color</span> <span m="2741420">that's</span> <span m="2741670">legal,</span> <span m="2742880">and</span> <span m="2743200">is</span> <span m="2743350">not</span> <span m="2743630">hooked</span> <span m="2743840">to</span> <span m="2743920">this</span> <span m="2744120">guy,</span> <span m="2744290">so</span> <span m="2744430">C1</span> <span m="2744870">is</span> <span m="2744980">legal.</span> <span m="2747110">What</span> <span m="2747280">color</span> <span m="2747560">do</span> <span m="2747650">I</span> <span m="2747820">give</span> <span m="2748095">here?</span> <span m="2749810">C2.</span> <span m="2752480">Then</span> <span m="2752790">I</span> <span m="2752870">color</span> <span m="2753160">this</span> <span m="2753370">one</span> <span m="2753550">next</span> <span m="2755710">C--</span> <span m="2757860">can't</span> <span m="2758115">do</span> <span m="2758370">C2,</span> <span m="2758660">can't</span> <span m="2758840">do</span> <span m="2758990">C1,</span> <span m="2759170">so</span> <span m="2759600">I</span> <span m="2759670">pick</span> <span m="2759890">C3.</span> <span m="2761460">And</span> <span m="2761600">then</span> <span m="2761740">I</span> <span m="2761780">get</span> <span m="2761930">to</span> <span m="2762010">6042</span> <span m="2762720">last,</span> <span m="2763330">and</span> <span m="2763450">I</span> <span m="2763500">can't</span> <span m="2763780">do</span> <span m="2763900">one,</span> <span m="2764220">two,</span> <span m="2764450">or</span> <span m="2764530">three,</span> <span m="2764810">so</span> <span m="2764930">I</span> <span m="2764990">do</span> <span m="2765130">four.</span> </p>
<p><span m="2767220">All</span> <span m="2767340">right</span> <span m="2767540">so</span> <span m="2767940">algorithm,</span> <span m="2768370">with</span> <span m="2768490">that</span> <span m="2768710">ordering,</span> <span m="2769150">gave</span> <span m="2769380">four</span> <span m="2769620">colors.</span> <span m="2770860">However</span> <span m="2771290">we</span> <span m="2771440">know</span> <span m="2771810">there's</span> <span m="2772020">a</span> <span m="2772090">way</span> <span m="2772370">to</span> <span m="2772500">do</span> <span m="2772680">a</span> <span m="2772750">different</span> <span m="2773170">ordering</span> <span m="2774380">that</span> <span m="2774970">gives</span> <span m="2775260">us</span> <span m="2776780">three</span> <span m="2777180">colors.</span> <span m="2779220">In</span> <span m="2779340">particular,</span> <span m="2780260">let's</span> <span m="2780370">see</span> <span m="2780490">if</span> <span m="2780600">we</span> <span m="2780700">do</span> <span m="2780840">this</span> <span m="2781180">what</span> <span m="2781260">happens</span> <span m="2781540">if</span> <span m="2781600">we</span> <span m="2781700">use</span> <span m="2781900">this</span> <span m="2782470">other</span> <span m="2782720">ordering.</span> <span m="2783150">Let</span> <span m="2783480">me</span> <span m="2783980">erase</span> <span m="2784235">these.</span> </p>
<p><span m="2786050">Say</span> <span m="2786290">that's</span> <span m="2786590">V1,</span> <span m="2788414">V2,</span> <span m="2789680">V3,</span> <span m="2790800">V4,</span> <span m="2792581">V5.</span> <span m="2793960">Now</span> <span m="2794150">I</span> <span m="2794250">get</span> <span m="2794460">C1,</span> <span m="2795740">this</span> <span m="2796010">will</span> <span m="2796170">be</span> <span m="2796330">C2,</span> <span m="2799790">C1.</span> <span m="2803920">What's</span> <span m="2804150">this</span> <span m="2804310">one</span> <span m="2804450">get?</span> <span m="2805670">C2.</span> <span m="2806320">Ah,</span> <span m="2807610">much</span> <span m="2807850">better.</span> <span m="2809350">C3.</span> <span m="2810800">So</span> <span m="2810990">different</span> <span m="2811430">orderings</span> <span m="2812250">result</span> <span m="2812710">in</span> <span m="2812860">different</span> <span m="2813250">numbers</span> <span m="2813570">of</span> <span m="2813670">colors</span> <span m="2814880">here.</span> <span m="2816380">So</span> <span m="2816720">the</span> <span m="2816820">whole</span> <span m="2817360">art</span> <span m="2817700">now</span> <span m="2817900">becomes</span> <span m="2819140">finding</span> <span m="2819840">a</span> <span m="2819940">clever</span> <span m="2820360">ordering.</span> <span m="2820850">And</span> <span m="2820990">so</span> <span m="2821290">many</span> <span m="2821600">people</span> <span m="2821940">have</span> <span m="2822000">already</span> <span m="2822250">had</span> <span m="2822590">good</span> <span m="2822930">ideas,</span> <span m="2823210">pick</span> <span m="2823430">the</span> <span m="2823520">largest</span> <span m="2823960">degree</span> <span m="2824260">nodes</span> <span m="2824610">first.</span> </p>
<p><span m="2825850">And</span> <span m="2826090">in</span> <span m="2826200">fact,</span> <span m="2828210">if</span> <span m="2828360">you</span> <span m="2828960">simulate</span> <span m="2829600">the</span> <span m="2829750">algorithm</span> <span m="2830080">on</span> <span m="2830170">lots</span> <span m="2830450">of</span> <span m="2830510">graphs,</span> <span m="2831740">you</span> <span m="2831920">do</span> <span m="2832070">better</span> <span m="2833340">on</span> <span m="2833610">average</span> <span m="2834420">when</span> <span m="2834700">you</span> <span m="2835150">color</span> <span m="2835370">the</span> <span m="2835590">larger</span> <span m="2835910">degree</span> <span m="2836140">nodes</span> <span m="2836420">first.</span> <span m="2837820">And</span> <span m="2837950">then</span> <span m="2838080">if</span> <span m="2838160">you</span> <span m="2838250">start</span> <span m="2838470">to</span> <span m="2838540">use</span> <span m="2838700">more</span> <span m="2838940">exotic</span> <span m="2839550">orderings,</span> <span m="2840850">you</span> <span m="2841010">can</span> <span m="2841180">do</span> <span m="2841370">even</span> <span m="2841620">better.</span> <span m="2843110">If</span> <span m="2843220">you</span> <span m="2843290">take</span> <span m="2843420">a</span> <span m="2843460">lot</span> <span m="2843700">of</span> <span m="2843750">graphs</span> <span m="2844130">that</span> <span m="2844230">are</span> <span m="2844340">out</span> <span m="2844520">there,</span> <span m="2844720">and</span> <span m="2844790">run</span> <span m="2844950">your</span> <span m="2845120">algorithm,</span> <span m="2845530">and</span> <span m="2845690">see</span> <span m="2845810">how</span> <span m="2845940">well</span> <span m="2846060">you</span> <span m="2846410">do,</span> <span m="2846550">you</span> <span m="2846920">do</span> <span m="2847040">better</span> <span m="2847360">with</span> <span m="2847560">more</span> <span m="2847760">sophisticated</span> <span m="2848410">orderings.</span> </p>
<p><span m="2849540">In</span> <span m="2849650">fact,</span> <span m="2851200">this</span> <span m="2851510">was</span> <span m="2852300">my</span> <span m="2853190">senior</span> <span m="2853540">thesis</span> <span m="2854340">back</span> <span m="2854550">when</span> <span m="2854660">I</span> <span m="2854730">was</span> <span m="2854860">undergraduate</span> <span m="2855480">student.</span> <span m="2856290">I</span> <span m="2856470">was</span> <span m="2856870">trying</span> <span m="2857160">to</span> <span m="2857220">figure</span> <span m="2857580">out</span> <span m="2858190">better</span> <span m="2858610">and</span> <span m="2858770">better</span> <span m="2859050">orderings</span> <span m="2859650">that</span> <span m="2859760">worked</span> <span m="2860200">for</span> <span m="2860390">graphs.</span> <span m="2861690">And</span> <span m="2862160">at</span> <span m="2862240">the</span> <span m="2862340">time</span> <span m="2862710">it</span> <span m="2862780">caused</span> <span m="2863120">a</span> <span m="2863160">bit</span> <span m="2863300">of</span> <span m="2863380">a</span> <span m="2863440">problem.</span> <span m="2864820">I</span> <span m="2865100">was</span> <span m="2865290">a</span> <span m="2865680">undergraduate</span> <span m="2866280">at</span> <span m="2866330">Princeton.</span> <span m="2867470">And</span> <span m="2867630">Princeton,</span> <span m="2868170">to</span> <span m="2868270">this</span> <span m="2868500">day</span> <span m="2868790">I</span> <span m="2868880">think,</span> <span m="2869120">still</span> <span m="2869330">has</span> <span m="2869540">exams</span> <span m="2870380">after</span> <span m="2871070">the</span> <span m="2871140">holidays,</span> <span m="2871450">the</span> <span m="2871760">Christmas</span> <span m="2872090">holidays,</span> <span m="2872365">New</span> <span m="2872640">Year's</span> <span m="2872880">holidays.</span> <span m="2874010">And</span> <span m="2874810">the</span> <span m="2874990">students</span> <span m="2876110">wanted</span> <span m="2876490">to</span> <span m="2876540">have</span> <span m="2876700">the</span> <span m="2876770">exams</span> <span m="2877180">before</span> <span m="2877430">Christmas,</span> <span m="2877810">because</span> <span m="2877980">they</span> <span m="2878030">hated</span> <span m="2878360">going</span> <span m="2878580">home</span> <span m="2878780">for</span> <span m="2878840">the</span> <span m="2878930">holiday,</span> <span m="2879770">and</span> <span m="2879930">then</span> <span m="2880030">you've</span> <span m="2880110">got</span> <span m="2880210">to</span> <span m="2880400">worry</span> <span m="2880870">about</span> <span m="2881200">your</span> <span m="2881360">exams</span> <span m="2881790">when</span> <span m="2881870">you</span> <span m="2881940">come</span> <span m="2882130">back.</span> <span m="2883270">And</span> <span m="2883500">the</span> <span m="2883570">faculty</span> <span m="2884180">said</span> <span m="2884500">no,</span> <span m="2885350">there's</span> <span m="2885560">no</span> <span m="2885860">way</span> <span m="2886050">to</span> <span m="2886120">get</span> <span m="2886290">them</span> <span m="2886420">all</span> <span m="2886600">compressed</span> <span m="2887160">into</span> <span m="2887340">a</span> <span m="2887400">small</span> <span m="2887670">number</span> <span m="2887880">of</span> <span m="2887950">days.</span> </p>
<p><span m="2889570">Now</span> <span m="2890220">I</span> <span m="2890610">wasn't</span> <span m="2890820">aware</span> <span m="2891120">of</span> <span m="2891220">all</span> <span m="2891350">that</span> <span m="2891520">of</span> <span m="2891570">the</span> <span m="2891670">time.</span> <span m="2892010">But</span> <span m="2892090">my</span> <span m="2892230">thesis</span> <span m="2892620">was</span> <span m="2892960">go</span> <span m="2893380">figure</span> <span m="2893660">out</span> <span m="2893810">good</span> <span m="2893980">ordering.</span> <span m="2894380">So</span> <span m="2894490">I</span> <span m="2894570">tried</span> <span m="2894850">lots</span> <span m="2895080">of</span> <span m="2895150">different</span> <span m="2895420">orderings.</span> <span m="2895880">And</span> <span m="2895990">I</span> <span m="2896200">tried</span> <span m="2896480">the</span> <span m="2896850">largest</span> <span m="2897180">degree</span> <span m="2897430">first,</span> <span m="2897850">and</span> <span m="2898530">recursive</span> <span m="2899100">versions</span> <span m="2899500">of</span> <span m="2899570">that</span> <span m="2899760">actually</span> <span m="2900070">worked</span> <span m="2900250">very</span> <span m="2900540">well.</span> <span m="2901730">And</span> <span m="2901910">then</span> <span m="2902750">tried</span> <span m="2903160">it</span> <span m="2903320">on</span> <span m="2903950">the</span> <span m="2904040">Princeton</span> <span m="2904400">exam</span> <span m="2904770">graph.</span> <span m="2905620">And</span> <span m="2905900">lo</span> <span m="2906090">and</span> <span m="2906180">behold,</span> <span m="2906740">you</span> <span m="2906840">could</span> <span m="2906870">actually</span> <span m="2907420">squish</span> <span m="2907820">it</span> <span m="2907930">down,</span> <span m="2908310">so</span> <span m="2908410">you</span> <span m="2908470">could</span> <span m="2908660">give</span> <span m="2908890">all</span> <span m="2909140">the</span> <span m="2909280">exams,</span> <span m="2909810">I</span> <span m="2909870">think</span> <span m="2910060">was,</span> <span m="2910200">4</span> <span m="2910380">and</span> <span m="2910560">1/2</span> <span m="2910740">days,</span> <span m="2911980">plenty</span> <span m="2912350">of</span> <span m="2912420">time</span> <span m="2912760">to</span> <span m="2912820">give</span> <span m="2912980">them</span> <span m="2913080">before</span> <span m="2913360">Christmas.</span> <span m="2913960">Which</span> <span m="2914440">caused</span> <span m="2914690">a</span> <span m="2915020">fair</span> <span m="2915070">of</span> <span m="2915360">scandal</span> <span m="2915860">at</span> <span m="2915920">the</span> <span m="2916020">time,</span> <span m="2916330">because</span> <span m="2916580">then</span> <span m="2917440">the</span> <span m="2917540">faculty</span> <span m="2917920">had</span> <span m="2918080">to</span> <span m="2918140">come</span> <span m="2918320">clean</span> <span m="2918670">that</span> <span m="2918750">they</span> <span m="2919220">just</span> <span m="2919400">didn't</span> <span m="2919450">want</span> <span m="2919840">to</span> <span m="2920110">bother</span> <span m="2920420">having</span> <span m="2920730">the</span> <span m="2921010">exams</span> <span m="2921315">before</span> <span m="2921620">Christmas.</span> </p>
<p><span m="2924180">Now</span> <span m="2924960">this</span> <span m="2925240">algorithm</span> <span m="2925740">is</span> <span m="2925850">an</span> <span m="2925920">example</span> <span m="2926470">of</span> <span m="2926540">what's</span> <span m="2926770">known</span> <span m="2926990">as</span> <span m="2927120">a</span> <span m="2927240">greedy</span> <span m="2927740">algorithm.</span> <span m="2928780">Now</span> <span m="2929260">in</span> <span m="2929350">a</span> <span m="2929390">greedy</span> <span m="2929820">algorithm</span> <span m="2931080">it's</span> <span m="2931310">always</span> <span m="2931840">simple.</span> <span m="2932930">You</span> <span m="2933060">just</span> <span m="2933540">go</span> <span m="2933690">one</span> <span m="2934000">step</span> <span m="2934280">after</span> <span m="2934500">the</span> <span m="2934580">next,</span> <span m="2935360">taking</span> <span m="2935760">the</span> <span m="2935850">best</span> <span m="2936310">you</span> <span m="2936410">can</span> <span m="2936580">do</span> <span m="2936870">at</span> <span m="2937100">each</span> <span m="2937430">stop.</span> <span m="2937760">You</span> <span m="2937980">never</span> <span m="2938320">go</span> <span m="2938470">back</span> <span m="2939170">and</span> <span m="2939290">try</span> <span m="2939450">to</span> <span m="2939540">make</span> <span m="2939710">things</span> <span m="2939940">better.</span> <span m="2940670">You</span> <span m="2940780">never</span> <span m="2941080">do</span> <span m="2941290">hill</span> <span m="2941640">climbing,</span> <span m="2942990">if</span> <span m="2943060">you're</span> <span m="2943170">familiar</span> <span m="2943480">with</span> <span m="2943580">that</span> <span m="2943770">term.</span> <span m="2943950">You</span> <span m="2944010">just</span> <span m="2944180">always</span> <span m="2944670">keep</span> <span m="2944890">it</span> <span m="2945000">simple,</span> <span m="2945670">one</span> <span m="2946120">thing</span> <span m="2946330">after</span> <span m="2946550">the</span> <span m="2946610">next,</span> <span m="2947010">very</span> <span m="2947340">fast.</span> <span m="2948830">Sometimes</span> <span m="2949085">it</span> <span m="2949340">works</span> <span m="2949590">great</span> <span m="2949850">in</span> <span m="2950160">practice.</span> <span m="2950770">Sometimes</span> <span m="2951175">it</span> <span m="2951790">doesn't.</span> <span m="2952570">But</span> <span m="2952680">it's</span> <span m="2952800">always</span> <span m="2953290">where</span> <span m="2953460">you</span> <span m="2953570">start,</span> <span m="2953980">some</span> <span m="2954150">simple</span> <span m="2954720">approach</span> <span m="2955160">like</span> <span m="2955360">this.</span> </p>
<p><span m="2956860">Now</span> <span m="2957980">this</span> <span m="2958270">algorithm</span> <span m="2958760">actually,</span> <span m="2960500">even</span> <span m="2960770">if</span> <span m="2960860">you</span> <span m="2960980">don't</span> <span m="2961360">try</span> <span m="2961570">to</span> <span m="2961680">monkey</span> <span m="2962040">with</span> <span m="2962180">the</span> <span m="2962310">ordering,</span> <span m="2962850">even</span> <span m="2963150">for</span> <span m="2963310">a</span> <span m="2963380">worst</span> <span m="2963690">case</span> <span m="2963950">ordering</span> <span m="2964490">of</span> <span m="2964620">the</span> <span m="2964690">nodes,</span> <span m="2965140">that</span> <span m="2965420">actually</span> <span m="2965720">does</span> <span m="2966020">pretty</span> <span m="2966340">good</span> <span m="2966990">for</span> <span m="2967170">a</span> <span m="2967220">lot</span> <span m="2967420">of</span> <span m="2967490">graphs.</span> <span m="2968930">And</span> <span m="2969080">in</span> <span m="2969170">fact,</span> <span m="2969610">it</span> <span m="2969690">does</span> <span m="2969940">really</span> <span m="2970260">well--</span> <span m="2970730">as</span> <span m="2970910">somebody</span> <span m="2971280">already</span> <span m="2971580">asked</span> <span m="2971880">about--</span> <span m="2973040">if</span> <span m="2973230">all</span> <span m="2973520">the</span> <span m="2973620">nodes</span> <span m="2974040">have</span> <span m="2974200">low</span> <span m="2974390">degree.</span> <span m="2975510">So</span> <span m="2975670">let's</span> <span m="2975890">state</span> <span m="2976110">that</span> <span m="2976390">as</span> <span m="2976540">a</span> <span m="2976620">theorem.</span> <span m="2977060">And</span> <span m="2977150">then</span> <span m="2977260">we're</span> <span m="2977330">going</span> <span m="2977410">to</span> <span m="2977460">prove</span> <span m="2977730">that.</span> </p>
<p><span m="2984329">So</span> <span m="2984810">if</span> <span m="2985080">every</span> <span m="2985370">node</span> <span m="2987010">in</span> <span m="2987090">a</span> <span m="2987150">graph</span> <span m="2987580">G</span> <span m="2989970">has</span> <span m="2990380">degree,</span> <span m="2993550">at</span> <span m="2993770">most,</span> <span m="2994330">d--</span> <span m="2994700">so</span> <span m="2995070">that's</span> <span m="2995240">the</span> <span m="2995320">biggest</span> <span m="2995680">degree</span> <span m="2995890">in</span> <span m="2995960">the</span> <span m="2996040">graph,</span> <span m="2996370">D--</span> <span m="2998120">then</span> <span m="2998550">this</span> <span m="2998760">basic</span> <span m="2999180">algorithm</span> <span m="3005570">uses,</span> <span m="3007640">at</span> <span m="3007840">most,</span> <span m="3010240">d</span> <span m="3010490">plus</span> <span m="3010870">1</span> <span m="3011400">colors</span> <span m="3014400">for</span> <span m="3014580">G.</span> <span m="3016480">No</span> <span m="3016650">matter</span> <span m="3016960">what</span> <span m="3017390">the</span> <span m="3017540">ordering</span> <span m="3017960">is,</span> <span m="3019750">you'll</span> <span m="3019920">never</span> <span m="3020240">do</span> <span m="3020400">worse</span> <span m="3020700">than</span> <span m="3020830">d</span> <span m="3021030">plus</span> <span m="3021280">1</span> <span m="3021470">colors.</span> </p>
<p><span m="3023490">So</span> <span m="3023700">what's</span> <span m="3023930">the</span> <span m="3024020">value</span> <span m="3024440">of</span> <span m="3024570">d</span> <span m="3024920">for</span> <span m="3025150">our</span> <span m="3025690">exam</span> <span m="3026030">graph</span> <span m="3026310">over</span> <span m="3026500">here?</span> <span m="3028840">d</span> <span m="3029040">is</span> <span m="3029230">3.</span> <span m="3030180">Every</span> <span m="3030580">node</span> <span m="3030880">has</span> <span m="3031090">degree,</span> <span m="3031410">at</span> <span m="3031490">most,</span> <span m="3031900">three.</span> <span m="3033240">And</span> <span m="3033500">so</span> <span m="3033670">it</span> <span m="3033780">says,</span> <span m="3034120">that</span> <span m="3034240">no</span> <span m="3034330">matter</span> <span m="3034570">what</span> <span m="3034860">ordering</span> <span m="3035220">you</span> <span m="3035410">picked</span> <span m="3035700">here,</span> <span m="3037120">you'd</span> <span m="3037230">get</span> <span m="3037440">at</span> <span m="3037500">most</span> <span m="3037760">four</span> <span m="3037950">colors.</span> <span m="3039520">Now</span> <span m="3039600">you</span> <span m="3039730">might</span> <span m="3039920">do</span> <span m="3040020">better.</span> <span m="3040350">In</span> <span m="3040430">fact,</span> <span m="3040680">we</span> <span m="3040780">found</span> <span m="3041060">an</span> <span m="3041130">ordering</span> <span m="3041420">that</span> <span m="3041510">got</span> <span m="3041720">three.</span> <span m="3042620">So</span> <span m="3042800">it's</span> <span m="3042910">possible</span> <span m="3043270">to</span> <span m="3043320">do</span> <span m="3043440">better.</span> </p>
<p><span m="3045940">So</span> <span m="3046100">let's</span> <span m="3046320">prove</span> <span m="3046640">this</span> <span m="3046840">fact</span> <span m="3047120">because</span> <span m="3047190">this</span> <span m="3048070">makes</span> <span m="3048360">a</span> <span m="3048450">difference.</span> <span m="3049040">Say</span> <span m="3049190">you</span> <span m="3049240">have</span> <span m="3049430">a</span> <span m="3049500">graph</span> <span m="3049850">with</span> <span m="3050690">hundreds</span> <span m="3051250">of</span> <span m="3051320">nodes.</span> <span m="3053260">But</span> <span m="3053410">every</span> <span m="3053600">node</span> <span m="3053850">has</span> <span m="3054070">degree,</span> <span m="3054370">at</span> <span m="3054440">most,</span> <span m="3054720">three.</span> <span m="3055716">Well</span> <span m="3056710">that</span> <span m="3056890">says</span> <span m="3057250">you</span> <span m="3057370">only</span> <span m="3057480">need</span> <span m="3057640">four</span> <span m="3057830">colors</span> <span m="3058320">even,</span> <span m="3058560">if</span> <span m="3058620">the</span> <span m="3058680">graph</span> <span m="3058950">has</span> <span m="3059120">1,000</span> <span m="3059670">nodes,</span> <span m="3060080">and</span> <span m="3060160">that's</span> <span m="3060470">very</span> <span m="3060710">useful.</span> <span m="3061840">So</span> <span m="3061880">in</span> <span m="3061940">that</span> <span m="3062090">kind</span> <span m="3062190">of</span> <span m="3062240">situation</span> <span m="3062555">it</span> <span m="3062870">does</span> <span m="3063110">very</span> <span m="3063390">well.</span> <span m="3064580">So</span> <span m="3064720">let's</span> <span m="3064900">prove</span> <span m="3065240">that.</span> <span m="3069570">Any</span> <span m="3069880">ideas</span> <span m="3070290">as</span> <span m="3070350">to</span> <span m="3070550">what</span> <span m="3070760">proof</span> <span m="3071030">technique</span> <span m="3071430">we're</span> <span m="3071540">going</span> <span m="3071650">to</span> <span m="3071690">use?</span> </p>
<p><span m="3072733">AUDIENCE: Invariant.</span> </p>
<p><span m="3074705">PROFESSOR: Invariant,</span> <span m="3076340">close.</span> <span m="3077120">Not</span> <span m="3077340">quite</span> <span m="3077580">an</span> <span m="3077670">invariant,</span> <span m="3078670">but</span> <span m="3078770">close.</span> </p>
<p><span m="3079560">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3082450">PROFESSOR: What?</span> </p>
<p><span m="3083620">AUDIENCE: Well</span> <span m="3084020">ordering</span> <span m="3084420">principle.</span> </p>
<p><span m="3084820">PROFESSOR: You</span> <span m="3084930">know</span> <span m="3085220">well</span> <span m="3085510">ordering</span> <span m="3085780">principle,</span> <span m="3086170">yeah,</span> <span m="3086420">we're</span> <span m="3086520">going</span> <span m="3086560">to</span> <span m="3086730">use</span> <span m="3086870">the</span> <span m="3087195">equivalent</span> <span m="3087520">version</span> <span m="3087930">of</span> <span m="3087980">that.</span> <span m="3088300">We're</span> <span m="3088430">going</span> <span m="3088540">to</span> <span m="3088580">use</span> <span m="3088820">induction.</span> <span m="3091410">If</span> <span m="3091590">you</span> <span m="3091690">like</span> <span m="3091860">well--</span> <span m="3092090">it's</span> <span m="3092260">equivalent</span> <span m="3092690">to</span> <span m="3092740">well</span> <span m="3092930">ordering.</span> <span m="3093190">If</span> <span m="3093360">you</span> <span m="3093570">like</span> <span m="3093780">well</span> <span m="3093890">ordering</span> <span m="3093980">you</span> <span m="3094110">could</span> <span m="3094240">do</span> <span m="3094380">it</span> <span m="3094460">that</span> <span m="3094580">way.</span> <span m="3094810">I</span> <span m="3094910">think</span> <span m="3095030">it's</span> <span m="3095210">easier</span> <span m="3095590">using</span> <span m="3095950">induction</span> <span m="3097040">here.</span> <span m="3098400">So</span> <span m="3098540">the</span> <span m="3098650">proof</span> <span m="3098980">is</span> <span m="3099120">by</span> <span m="3099350">induction.</span> </p>
<p><span m="3105110">All</span> <span m="3105350">right</span> <span m="3105640">so</span> <span m="3105740">the</span> <span m="3105840">first</span> <span m="3106330">thing</span> <span m="3106460">we</span> <span m="3106570">need</span> <span m="3106750">is</span> <span m="3106850">an</span> <span m="3106940">induction</span> <span m="3107360">hypothesis.</span> <span m="3110950">Any</span> <span m="3111190">thoughts</span> <span m="3111630">about</span> <span m="3111890">what</span> <span m="3112040">the</span> <span m="3112140">induction</span> <span m="3112460">hypothesis</span> <span m="3113070">should</span> <span m="3113230">be?</span> <span m="3114450">Yeah?</span> </p>
<p><span m="3115202">AUDIENCE: If</span> <span m="3115664">you</span> <span m="3116126">have</span> <span m="3116588">a</span> <span m="3117050">graph</span> <span m="3117512">with</span> <span m="3117974">n</span> <span m="3118440">nodes</span> <span m="3118725">then</span> <span m="3121270">where</span> <span m="3121722">the</span> <span m="3122630">degree</span> <span m="3122890">of</span> <span m="3123080">any</span> <span m="3123370">nodes</span> <span m="3123720">is</span> <span m="3124206">less</span> <span m="3124692">than</span> <span m="3125178">[INAUDIBLE]</span> <span m="3126150">then</span> <span m="3126636">you</span> <span m="3127122">can</span> <span m="3127608">do</span> <span m="3128094">it.</span> </p>
<p><span m="3128580">PROFESSOR: That's</span> <span m="3128700">great.</span> <span m="3129090">You're</span> <span m="3129560">going</span> <span m="3130055">to</span> <span m="3130550">do</span> <span m="3130630">really</span> <span m="3131030">well</span> <span m="3131170">on</span> <span m="3131540">the</span> <span m="3131830">midterm,</span> <span m="3132030">because</span> <span m="3132280">you</span> <span m="3132600">put</span> <span m="3133005">an</span> <span m="3133410">n</span> <span m="3133815">into</span> <span m="3134220">this</span> <span m="3134300">thing,</span> <span m="3134450">but</span> <span m="3134730">there's</span> <span m="3134960">not</span> <span m="3135120">an</span> <span m="3135200">n</span> <span m="3135610">here</span> <span m="3135980">to</span> <span m="3136290">start.</span> <span m="3136570">What</span> <span m="3136690">are</span> <span m="3136770">most</span> <span m="3137050">people</span> <span m="3137330">going</span> <span m="3137770">to</span> <span m="3138210">do--</span> <span m="3138320">we</span> <span m="3138490">used</span> <span m="3138950">to</span> <span m="3139340">ask</span> <span m="3139500">this</span> <span m="3139800">actually.</span> <span m="3140210">We</span> <span m="3140310">asked</span> <span m="3140380">this</span> <span m="3140820">once</span> <span m="3141260">on</span> <span m="3141550">a</span> <span m="3141770">test</span> <span m="3142040">many</span> <span m="3142310">years</span> <span m="3142620">ago,</span> <span m="3142830">and</span> <span m="3142900">it</span> <span m="3143050">was</span> <span m="3143160">an</span> <span m="3143380">utter</span> <span m="3143660">disaster,</span> <span m="3143940">because</span> <span m="3144110">did</span> <span m="3144360">everybody</span> <span m="3144760">do?</span> <span m="3145820">May</span> <span m="3145920">be</span> <span m="3146080">one</span> <span m="3146400">student,</span> <span m="3146720">or</span> <span m="3146790">two,</span> <span m="3147080">put</span> <span m="3147280">an</span> <span m="3147450">n</span> <span m="3147780">into</span> <span m="3148120">there.</span> <span m="3148980">But</span> <span m="3149090">what's</span> <span m="3149300">the</span> <span m="3149400">naturally</span> <span m="3149800">thing</span> <span m="3150060">to</span> <span m="3150140">do</span> <span m="3150390">to</span> <span m="3150520">induct</span> <span m="3150890">on</span> <span m="3151050">here</span> <span m="3151210">when</span> <span m="3151290">you</span> <span m="3151400">look</span> <span m="3151650">at</span> <span m="3151730">this</span> <span m="3152180">statement?</span> <span m="3154560">You're</span> <span m="3154710">going</span> <span m="3154810">to</span> <span m="3154850">induct</span> <span m="3155240">on</span> <span m="3155410">d,</span> <span m="3156630">because</span> <span m="3156780">the</span> <span m="3156860">first</span> <span m="3157150">thing</span> <span m="3157280">you</span> <span m="3157380">do</span> <span m="3157760">is</span> <span m="3157850">you</span> <span m="3157950">make</span> <span m="3158160">this</span> <span m="3158550">be</span> <span m="3158720">your</span> <span m="3159040">induction</span> <span m="3159360">hypothesis.</span> <span m="3159680">There's</span> <span m="3159840">only</span> <span m="3160150">one</span> <span m="3160450">thing</span> <span m="3160640">to</span> <span m="3160690">use,</span> <span m="3161710">so</span> <span m="3161850">you're</span> <span m="3161950">going</span> <span m="3162050">to</span> <span m="3162100">have</span> <span m="3162540">your</span> <span m="3163160">predicate</span> <span m="3163640">be</span> <span m="3163990">p</span> <span m="3164340">of</span> <span m="3164690">d,</span> <span m="3164910">and</span> <span m="3165150">it's</span> <span m="3165280">going</span> <span m="3165320">to</span> <span m="3165430">be</span> <span m="3165780">that.</span> </p>
<p><span m="3167280">Now</span> <span m="3167780">It</span> <span m="3167870">didn't</span> <span m="3168070">occur</span> <span m="3168410">to</span> <span m="3168530">us</span> <span m="3168710">that's</span> <span m="3168840">what</span> <span m="3169170">everybody</span> <span m="3169270">was</span> <span m="3169350">going</span> <span m="3169420">to</span> <span m="3169590">do,</span> <span m="3169680">but</span> <span m="3169870">it</span> <span m="3170170">should</span> <span m="3170430">have.</span> <span m="3170560">They</span> <span m="3170730">all</span> <span m="3170910">did</span> <span m="3171330">that</span> <span m="3171740">and</span> <span m="3171800">it</span> <span m="3171940">was</span> <span m="3172010">a</span> <span m="3172350">disaster.</span> <span m="3173220">Because</span> <span m="3173520">if</span> <span m="3173630">you</span> <span m="3173760">do</span> <span m="3174110">this,</span> <span m="3175480">well</span> <span m="3175830">you've</span> <span m="3176010">got</span> <span m="3176200">to</span> <span m="3176270">take</span> <span m="3176470">a</span> <span m="3176520">graph</span> <span m="3177010">with</span> <span m="3177150">maximum</span> <span m="3177590">degree</span> <span m="3177950">d,</span> <span m="3179460">or</span> <span m="3179580">d</span> <span m="3179750">plus</span> <span m="3180000">1</span> <span m="3180280">in</span> <span m="3180360">the</span> <span m="3180430">inductive</span> <span m="3180780">step,</span> <span m="3181890">pull</span> <span m="3182220">out</span> <span m="3182390">all</span> <span m="3182780">the</span> <span m="3182880">nodes</span> <span m="3183200">with</span> <span m="3183320">degree</span> <span m="3183620">d</span> <span m="3183810">plus</span> <span m="3184090">1</span> <span m="3185640">to</span> <span m="3185750">get</span> <span m="3185910">a</span> <span m="3185960">graph</span> <span m="3186330">with</span> <span m="3186420">now</span> <span m="3186590">degree</span> <span m="3186910">d.</span> <span m="3187210">And</span> <span m="3187270">that's</span> <span m="3187450">a</span> <span m="3187490">mess.</span> <span m="3187770">You</span> <span m="3188050">just</span> <span m="3188270">pulled</span> <span m="3188360">out</span> <span m="3188410">a</span> <span m="3188630">lot</span> <span m="3188690">of</span> <span m="3189130">nodes,</span> <span m="3189500">potentially.</span> <span m="3190410">Color</span> <span m="3190690">that</span> <span m="3190920">in</span> <span m="3191170">d</span> <span m="3191380">plus</span> <span m="3191560">1</span> <span m="3191990">colors,</span> <span m="3192220">now</span> <span m="3192380">put</span> <span m="3192540">all</span> <span m="3192810">that</span> <span m="3193020">junk</span> <span m="3193340">back</span> <span m="3193640">in.</span> <span m="3194650">And</span> <span m="3194850">say</span> <span m="3195140">only</span> <span m="3195390">used</span> <span m="3195540">one</span> <span m="3195710">more</span> <span m="3195910">color.</span> <span m="3197180">Nightmare.</span> <span m="3198520">And</span> <span m="3198590">these</span> <span m="3198760">were</span> <span m="3198830">MIT</span> <span m="3199260">students</span> <span m="3199540">under</span> <span m="3199720">pressure.</span> <span m="3201300">It</span> <span m="3201560">was</span> <span m="3201850">a</span> <span m="3201910">nightmare.</span> </p>
<p><span m="3203330">So</span> <span m="3203350">that</span> <span m="3203580">does</span> <span m="3203730">not</span> <span m="3203990">work.</span> <span m="3204410">And</span> <span m="3204640">in</span> <span m="3204700">fact,</span> <span m="3205920">we</span> <span m="3206040">will</span> <span m="3206320">ask</span> <span m="3206630">an</span> <span m="3206700">induction</span> <span m="3207080">question</span> <span m="3207410">on</span> <span m="3207500">graphs</span> <span m="3207850">on</span> <span m="3207950">every</span> <span m="3208180">test</span> <span m="3208450">you</span> <span m="3208540">take</span> <span m="3208830">in</span> <span m="3208990">this</span> <span m="3209450">course.</span> <span m="3209960">It</span> <span m="3210070">will</span> <span m="3210230">happen.</span> <span m="3211190">And</span> <span m="3211790">so</span> <span m="3212460">usually,</span> <span m="3214130">with</span> <span m="3214310">induction,</span> <span m="3214760">you</span> <span m="3214850">take</span> <span m="3215090">this</span> <span m="3215330">as</span> <span m="3215400">your</span> <span m="3215520">induction</span> <span m="3215830">hypothesis.</span> <span m="3216580">With</span> <span m="3216720">graphs,</span> <span m="3217150">you</span> <span m="3217230">have</span> <span m="3217340">to</span> <span m="3217400">be</span> <span m="3217490">careful.</span> <span m="3218650">And</span> <span m="3220040">worst</span> <span m="3220270">part</span> <span m="3220430">about</span> <span m="3220620">this</span> <span m="3220840">is</span> <span m="3221250">we</span> <span m="3221370">tell</span> <span m="3221570">people</span> <span m="3221830">when</span> <span m="3221970">this</span> <span m="3222150">doesn't</span> <span m="3222440">work,</span> <span m="3223300">use</span> <span m="3223560">a</span> <span m="3223630">stronger</span> <span m="3224160">induction</span> <span m="3224490">hypothesis.</span> <span m="3225950">So</span> <span m="3226410">students</span> <span m="3226700">tried</span> <span m="3226950">to</span> <span m="3227010">make</span> <span m="3227190">a</span> <span m="3227250">stronger,</span> <span m="3228280">but</span> <span m="3228430">they're</span> <span m="3228570">still</span> <span m="3228830">stuck</span> <span m="3229090">on</span> <span m="3229220">d,</span> <span m="3229560">and</span> <span m="3229660">it</span> <span m="3229880">was</span> <span m="3230325">still</span> <span m="3230770">a</span> <span m="3231146">disaster.</span> </p>
<p><span m="3231900">With</span> <span m="3232070">graphs,</span> <span m="3232580">you</span> <span m="3232680">do</span> <span m="3232820">something</span> <span m="3233180">different.</span> <span m="3233930">And</span> <span m="3234100">the</span> <span m="3234180">first</span> <span m="3234530">thing</span> <span m="3234650">you</span> <span m="3234730">do</span> <span m="3234870">with</span> <span m="3234960">a</span> <span m="3235030">graph,</span> <span m="3235470">usually,</span> <span m="3236240">is</span> <span m="3236690">put</span> <span m="3236940">n</span> <span m="3237170">in</span> <span m="3237300">here.</span> <span m="3238400">And</span> <span m="3238560">if</span> <span m="3238620">it</span> <span m="3238700">doesn't</span> <span m="3238990">work</span> <span m="3239190">with</span> <span m="3239350">n,</span> <span m="3239540">the</span> <span m="3239600">number</span> <span m="3239860">of</span> <span m="3239930">nodes,</span> <span m="3240510">you</span> <span m="3240670">put</span> <span m="3240880">in</span> <span m="3241410">e</span> <span m="3241650">the</span> <span m="3241740">number</span> <span m="3241980">of</span> <span m="3242050">edges.</span> <span m="3243410">And</span> <span m="3243550">induct</span> <span m="3243960">on</span> <span m="3244100">that.</span> <span m="3245120">And</span> <span m="3245290">so</span> <span m="3245400">what</span> <span m="3245520">you</span> <span m="3245680">said</span> <span m="3245880">is</span> <span m="3245980">exactly</span> <span m="3246420">the</span> <span m="3246520">right</span> <span m="3246720">thing</span> <span m="3246880">to</span> <span m="3246950">do.</span> <span m="3248460">Don't</span> <span m="3248750">do</span> <span m="3248880">this,</span> <span m="3249550">or</span> <span m="3249700">least</span> <span m="3249960">don't</span> <span m="3250260">spend</span> <span m="3250500">too</span> <span m="3250580">much</span> <span m="3250740">time</span> <span m="3251000">on</span> <span m="3251130">it.</span> <span m="3251270">Pretty</span> <span m="3251500">quickly</span> <span m="3251870">try</span> <span m="3252130">this.</span> <span m="3253360">If</span> <span m="3253520">every</span> <span m="3253940">end</span> <span m="3254230">node</span> <span m="3254520">graph--</span> <span m="3255790">if</span> <span m="3255920">every</span> <span m="3256360">node</span> <span m="3257030">in</span> <span m="3257310">an</span> <span m="3257500">n</span> <span m="3257970">node</span> <span m="3258210">graph</span> <span m="3258530">G</span> <span m="3261740">has</span> <span m="3262180">degree</span> <span m="3262460">at</span> <span m="3262570">most</span> <span m="3262770">degree,</span> <span m="3263840">then</span> <span m="3264110">the</span> <span m="3264190">basic</span> <span m="3264550">algorithm</span> <span m="3264850">uses</span> <span m="3265200">at</span> <span m="3265250">most</span> <span m="3265610">d,</span> <span m="3265840">plus</span> <span m="3266000">one</span> <span m="3266265">colors.</span> <span m="3267070">And</span> <span m="3267310">now</span> <span m="3267620">you</span> <span m="3267780">induct</span> <span m="3267920">on</span> <span m="3268750">n.</span> <span m="3270350">And</span> <span m="3270550">almost</span> <span m="3270950">always</span> <span m="3271200">on</span> <span m="3271250">graphs,</span> <span m="3271740">that's</span> <span m="3271990">the</span> <span m="3272140">first</span> <span m="3272520">thing</span> <span m="3272650">to</span> <span m="3272710">try.</span> <span m="3273100">Even</span> <span m="3273340">if</span> <span m="3273410">it's</span> <span m="3273580">not</span> <span m="3274840">in</span> <span m="3275000">your</span> <span m="3275200">theorem</span> <span m="3275520">statement.</span> <span m="3277270">Any</span> <span m="3277390">questions</span> <span m="3279380">about</span> <span m="3279650">that?</span> </p>
<p><span m="3286940">Well</span> <span m="3287330">let's</span> <span m="3287650">start</span> <span m="3287910">with</span> <span m="3288020">this,</span> <span m="3288310">and</span> <span m="3289390">see</span> <span m="3289570">if</span> <span m="3289650">we</span> <span m="3289740">can</span> <span m="3289830">make</span> <span m="3290050">this</span> <span m="3290260">one</span> <span m="3290758">work.</span> <span m="3302230">So</span> <span m="3302390">what's</span> <span m="3302610">the</span> <span m="3302940">next</span> <span m="3303190">step</span> <span m="3303490">in</span> <span m="3303620">our</span> <span m="3303710">proof?</span> <span m="3304140">What</span> <span m="3304440">do</span> <span m="3304590">we</span> <span m="3304670">got</span> <span m="3304790">to</span> <span m="3304850">do?</span> <span m="3306560">Base</span> <span m="3306880">case.</span> <span m="3311890">And</span> <span m="3312070">the</span> <span m="3312140">base</span> <span m="3312470">case</span> <span m="3312850">will</span> <span m="3313000">be,</span> <span m="3313740">not</span> <span m="3313980">n</span> <span m="3314120">equals</span> <span m="3314420">0,</span> <span m="3314750">because</span> <span m="3314910">we</span> <span m="3315000">can't</span> <span m="3315250">have</span> <span m="3315400">a</span> <span m="3315420">zero</span> <span m="3315730">node</span> <span m="3315890">graph,</span> <span m="3316330">but</span> <span m="3316990">n</span> <span m="3317180">equals</span> <span m="3317450">1.</span> <span m="3319300">And</span> <span m="3319930">how</span> <span m="3320100">many</span> <span m="3320330">edges</span> <span m="3320600">do</span> <span m="3320680">we</span> <span m="3320780">have?</span> <span m="3322940">Zero.</span> <span m="3323440">If</span> <span m="3323510">there's</span> <span m="3323670">one</span> <span m="3323950">node,</span> <span m="3324340">we</span> <span m="3324440">don't</span> <span m="3324610">allow</span> <span m="3324820">loops,</span> <span m="3325090">so</span> <span m="3325340">it's</span> <span m="3325940">zero</span> <span m="3326350">edges,</span> <span m="3329100">which</span> <span m="3329610">means</span> <span m="3330040">that</span> <span m="3330790">the</span> <span m="3330930">degree</span> <span m="3332520">of</span> <span m="3332660">our</span> <span m="3332820">graph</span> <span m="3333140">has</span> <span m="3333420">to</span> <span m="3333520">be</span> <span m="3335790">zero.</span> <span m="3336380">There's</span> <span m="3336540">no</span> <span m="3336740">edges.</span> <span m="3337780">And</span> <span m="3338300">of</span> <span m="3338380">course</span> <span m="3338780">there's</span> <span m="3338910">only</span> <span m="3339100">one</span> <span m="3339370">node,</span> <span m="3339900">so</span> <span m="3340150">one</span> <span m="3340390">color</span> <span m="3342560">is</span> <span m="3342700">going</span> <span m="3342830">to</span> <span m="3342940">work,</span> <span m="3343850">and</span> <span m="3343980">that</span> <span m="3344140">happens</span> <span m="3344520">to</span> <span m="3344590">equal</span> <span m="3345530">d</span> <span m="3345720">plus</span> <span m="3346000">1.</span> </p>
<p><span m="3349460">All</span> <span m="3349560">right,</span> <span m="3349850">so</span> <span m="3350900">the</span> <span m="3351040">base</span> <span m="3351330">case</span> <span m="3351570">is</span> <span m="3351670">true.</span> <span m="3352110">For</span> <span m="3352270">one</span> <span m="3352660">node</span> <span m="3352950">graphs,</span> <span m="3354440">you</span> <span m="3354650">can</span> <span m="3354800">always</span> <span m="3355110">use</span> <span m="3355340">d</span> <span m="3355520">plus</span> <span m="3355760">1</span> <span m="3355940">colors,</span> <span m="3356695">where</span> <span m="3357010">d</span> <span m="3357430">is</span> <span m="3357500">the</span> <span m="3358110">max</span> <span m="3358560">degree.</span> </p>
<p><span m="3359010">All</span> <span m="3359460">right,</span> <span m="3359910">next</span> <span m="3360170">we</span> <span m="3360240">have</span> <span m="3360420">the</span> <span m="3360500">inductive</span> <span m="3360940">step.</span> <span m="3365500">So</span> <span m="3365670">here</span> <span m="3365940">we</span> <span m="3366090">assume</span> <span m="3366920">P</span> <span m="3367295">n</span> <span m="3368770">is</span> <span m="3368920">true</span> <span m="3373620">for</span> <span m="3374040">the</span> <span m="3374160">induction.</span> <span m="3380330">And</span> <span m="3380600">now</span> <span m="3380850">we</span> <span m="3381010">look</span> <span m="3381230">at</span> <span m="3381320">an</span> <span m="3381470">n</span> <span m="3381710">plus</span> <span m="3382000">1</span> <span m="3382270">node</span> <span m="3382480">graph</span> <span m="3383130">to</span> <span m="3383590">show</span> <span m="3383830">P</span> <span m="3384140">n</span> <span m="3384330">plus</span> <span m="3384490">1</span> <span m="3384570">is</span> <span m="3384820">true.</span> <span m="3386620">So</span> <span m="3386640">we</span> <span m="3386710">let</span> <span m="3386910">G</span> <span m="3389980">be</span> <span m="3390500">any</span> <span m="3392436">N</span> <span m="3392920">plus</span> <span m="3393230">1</span> <span m="3393450">node</span> <span m="3393690">graph.</span> <span m="3395240">We</span> <span m="3395650">got</span> <span m="3396060">to</span> <span m="3396130">show</span> <span m="3396950">you</span> <span m="3397090">can</span> <span m="3397270">color</span> <span m="3397630">it</span> <span m="3397770">in</span> <span m="3397930">d</span> <span m="3398130">plus</span> <span m="3398300">1</span> <span m="3398710">colors.</span> <span m="3401810">And</span> <span m="3401900">let's</span> <span m="3402240">let</span> <span m="3403070">d</span> <span m="3403470">be</span> <span m="3403580">the</span> <span m="3403680">max</span> <span m="3403890">degree,</span> <span m="3408160">the</span> <span m="3408260">largest</span> <span m="3408790">degree</span> <span m="3409320">in</span> <span m="3409760">G.</span> </p>
<p><span m="3416020">We've</span> <span m="3416140">got</span> <span m="3416310">to</span> <span m="3416370">show</span> <span m="3416700">we</span> <span m="3416800">can</span> <span m="3416950">color</span> <span m="3417440">it</span> <span m="3417700">in</span> <span m="3417970">d</span> <span m="3418160">plus</span> <span m="3418410">1</span> <span m="3418600">colors.</span> <span m="3421180">Well</span> <span m="3421340">the</span> <span m="3421430">basic</span> <span m="3421800">algorithm,</span> <span m="3422270">let's</span> <span m="3422500">say.</span> <span m="3423120">First</span> <span m="3423330">thing</span> <span m="3423410">we</span> <span m="3423490">do</span> <span m="3423610">is</span> <span m="3423710">we</span> <span m="3423780">order</span> <span m="3424050">the</span> <span m="3424160">nodes</span> <span m="3425680">in</span> <span m="3425850">an</span> <span m="3425980">arbitrary</span> <span m="3426467">order.</span> <span m="3430363">And</span> <span m="3430850">we're</span> <span m="3431010">going</span> <span m="3431090">to</span> <span m="3431150">show</span> <span m="3431590">whatever</span> <span m="3432130">order</span> <span m="3432350">you</span> <span m="3432490">pick</span> <span m="3432730">is</span> <span m="3433215">OK.</span> </p>
<p></p>
<p><span m="3442920">All</span> <span m="3443090">right</span> <span m="3443290">so</span> <span m="3447490">what</span> <span m="3447710">are</span> <span m="3447760">the</span> <span m="3447870">nodes?</span> <span m="3458740">Anyway</span> <span m="3459140">at</span> <span m="3459230">all.</span> <span m="3461200">Now</span> <span m="3461660">how</span> <span m="3461920">am</span> <span m="3462180">I</span> <span m="3462520">going</span> <span m="3462730">to</span> <span m="3462910">use</span> <span m="3463260">the</span> <span m="3463400">induction</span> <span m="3463870">hypothesis?</span> <span m="3465690">I</span> <span m="3465820">know,</span> <span m="3466680">I</span> <span m="3466780">can</span> <span m="3466930">assume,</span> <span m="3468140">the</span> <span m="3468270">for</span> <span m="3468410">any</span> <span m="3468670">N</span> <span m="3468930">node</span> <span m="3469190">graph</span> <span m="3469900">I</span> <span m="3470030">can</span> <span m="3470200">color</span> <span m="3470620">it</span> <span m="3471820">in</span> <span m="3471970">the</span> <span m="3472050">max</span> <span m="3472380">degree</span> <span m="3472670">plus</span> <span m="3472940">1</span> <span m="3473420">colors.</span> <span m="3474550">How</span> <span m="3474810">am</span> <span m="3474910">I</span> <span m="3474950">going</span> <span m="3475080">to</span> <span m="3475130">use</span> <span m="3475370">that</span> <span m="3475590">to</span> <span m="3475670">help</span> <span m="3475900">me</span> <span m="3476020">color</span> <span m="3476390">G</span> <span m="3476710">here,</span> <span m="3476940">the</span> <span m="3477070">n</span> <span m="3477180">plus</span> <span m="3477400">1</span> <span m="3477740">node</span> <span m="3478000">graph?</span> <span m="3478260">Any</span> <span m="3478420">thoughts?</span> <span m="3480950">Yeah?</span> </p>
<p><span m="3481190">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3483860">PROFESSOR: Yeah,</span> <span m="3484830">let's</span> <span m="3485280">create</span> <span m="3485760">an</span> <span m="3486130">n</span> <span m="3486330">node</span> <span m="3486625">graph</span> <span m="3486920">by</span> <span m="3487870">looking</span> <span m="3488150">at</span> <span m="3488250">these</span> <span m="3488520">nodes,</span> <span m="3488880">and</span> <span m="3489020">taking</span> <span m="3489410">this</span> <span m="3489570">one</span> <span m="3489750">out</span> <span m="3490040">of</span> <span m="3490080">the</span> <span m="3490150">time</span> <span m="3490450">being.</span> <span m="3492310">Remove</span> <span m="3492705">the</span> <span m="3493100">last</span> <span m="3493190">V</span> <span m="3493650">n</span> <span m="3493690">plus</span> <span m="3494180">1</span> <span m="3494350">node</span> <span m="3494710">in</span> <span m="3495010">the</span> <span m="3495460">order.</span> <span m="3495910">That</span> <span m="3496130">leaves</span> <span m="3496410">an</span> <span m="3496510">n</span> <span m="3496880">node</span> <span m="3497250">graph.</span> <span m="3499400">So</span> <span m="3500330">let's</span> <span m="3500450">write</span> <span m="3500640">that</span> <span m="3500840">down.</span> </p>
<p><span m="3502310">We</span> <span m="3502450">remove</span> <span m="3504430">the</span> <span m="3504650">n</span> <span m="3504810">plus</span> <span m="3505110">1</span> <span m="3506110">from</span> <span m="3506670">G.</span> <span m="3509650">And</span> <span m="3509770">that</span> <span m="3509940">creates</span> <span m="3510350">a</span> <span m="3510400">new</span> <span m="3510600">graph,</span> <span m="3513120">call</span> <span m="3513420">it</span> <span m="3513550">G</span> <span m="3513880">prime</span> <span m="3515450">with</span> <span m="3516100">vertices,</span> <span m="3516660">V</span> <span m="3516860">prime</span> <span m="3517190">and</span> <span m="3517290">edges,</span> <span m="3517360">E</span> <span m="3517700">prime.</span> <span m="3518530">So</span> <span m="3518910">we</span> <span m="3519040">create</span> <span m="3519270">a</span> <span m="3519340">new</span> <span m="3519540">graph</span> <span m="3519960">by</span> <span m="3520060">removing</span> <span m="3520430">that</span> <span m="3520650">node.</span> <span m="3523640">And</span> <span m="3523970">we</span> <span m="3524080">remove</span> <span m="3524360">all</span> <span m="3524530">the</span> <span m="3524680">edges</span> <span m="3525440">tied</span> <span m="3525730">to</span> <span m="3525770">that</span> <span m="3525990">node.</span> </p>
<p><span m="3526240">So</span> <span m="3526430">for</span> <span m="3526550">example</span> <span m="3527010">over</span> <span m="3527180">here,</span> <span m="3529520">the</span> <span m="3529620">last</span> <span m="3529980">node</span> <span m="3530360">was</span> <span m="3530690">6042,</span> <span m="3531440">so</span> <span m="3531570">we</span> <span m="3531670">take</span> <span m="3531960">out</span> <span m="3532160">6042,</span> <span m="3534190">and</span> <span m="3534380">all</span> <span m="3534530">these</span> <span m="3534740">edges.</span> <span m="3536150">And</span> <span m="3536440">this</span> <span m="3536650">is</span> <span m="3536780">a</span> <span m="3536840">graph</span> <span m="3537260">that</span> <span m="3537724">we're</span> <span m="3538188">left</span> <span m="3538652">with.</span> <span m="3542370">That</span> <span m="3542630">graph</span> <span m="3543230">has</span> <span m="3543920">n</span> <span m="3544120">nodes.</span> <span m="3546190">What's</span> <span m="3546680">the</span> <span m="3546790">maximum</span> <span m="3547340">degree</span> <span m="3547770">in</span> <span m="3547890">G</span> <span m="3548170">prime?</span> <span m="3551910">When</span> <span m="3552060">I</span> <span m="3552140">pull</span> <span m="3552500">out</span> <span m="3552610">a</span> <span m="3552660">node,</span> <span m="3555030">can</span> <span m="3555130">the</span> <span m="3555220">degree</span> <span m="3555780">of</span> <span m="3555850">any</span> <span m="3556010">node</span> <span m="3556240">go</span> <span m="3556460">up?</span> <span m="3558054">No,</span> <span m="3558930">I'm</span> <span m="3559190">just</span> <span m="3559370">taking</span> <span m="3559640">stuff</span> <span m="3559970">out.</span> </p>
<p><span m="3561320">So</span> <span m="3562740">I</span> <span m="3562880">know</span> <span m="3563200">that</span> <span m="3563500">G</span> <span m="3563800">prime</span> <span m="3565020">has</span> <span m="3565490">maximum</span> <span m="3565970">degree,</span> <span m="3566370">at</span> <span m="3566480">most,</span> <span m="3567030">d.</span> <span m="3567930">The</span> <span m="3568830">degree</span> <span m="3569240">didn't</span> <span m="3569450">go</span> <span m="3569550">up</span> <span m="3569700">of</span> <span m="3569800">any</span> <span m="3569970">node.</span> <span m="3575080">Might</span> <span m="3575560">have</span> <span m="3575740">gone</span> <span m="3575960">down,</span> <span m="3577100">but</span> <span m="3577230">it</span> <span m="3577300">didn't</span> <span m="3577510">go</span> <span m="3577680">up.</span> <span m="3578430">So</span> <span m="3578570">G</span> <span m="3578810">prime</span> <span m="3579190">has</span> <span m="3579400">max</span> <span m="3579750">degree,</span> <span m="3580340">at</span> <span m="3580550">most,</span> <span m="3581200">d,</span> <span m="3582454">and</span> <span m="3583290">it</span> <span m="3583350">has</span> <span m="3583600">n</span> <span m="3583790">nodes.</span> <span m="3587440">So</span> <span m="3587520">we</span> <span m="3587640">can</span> <span m="3587780">use</span> <span m="3588100">the</span> <span m="3588210">induction</span> <span m="3588590">hypothesis</span> <span m="3589690">P</span> <span m="3590065">n.</span> <span m="3593360">It</span> <span m="3593520">says</span> <span m="3593775">that</span> <span m="3594130">the</span> <span m="3594210">basic</span> <span m="3594690">algorithm</span> <span m="3599310">uses</span> <span m="3600370">d</span> <span m="3600680">plus</span> <span m="3600990">1,</span> <span m="3601320">at</span> <span m="3601410">most,</span> <span m="3602390">d</span> <span m="3602560">plus</span> <span m="3602840">1</span> <span m="3604330">colors</span> <span m="3606410">for</span> <span m="3606850">nodes</span> <span m="3607320">V1</span> <span m="3608610">to</span> <span m="3608910">V</span> <span m="3609280">n.</span> <span m="3613104">Any</span> <span m="3613590">questions</span> <span m="3613720">about</span> <span m="3613780">that?</span> </p>
<p><span m="3617140">So</span> <span m="3617530">if</span> <span m="3618320">this</span> <span m="3618530">were</span> <span m="3618630">the</span> <span m="3618790">n</span> <span m="3618910">plus</span> <span m="3619210">first</span> <span m="3619500">node,</span> <span m="3620830">last</span> <span m="3621190">node</span> <span m="3621340">in</span> <span m="3621400">the</span> <span m="3621510">ordering</span> <span m="3622140">take</span> <span m="3622380">it</span> <span m="3622480">out.</span> <span m="3624090">The</span> <span m="3624190">basic</span> <span m="3624640">algorithm</span> <span m="3625140">now,</span> <span m="3625480">take</span> <span m="3625720">the</span> <span m="3625800">same</span> <span m="3626080">order</span> <span m="3626310">here,</span> <span m="3626490">V1,</span> <span m="3626650">V2,</span> <span m="3627140">V3,</span> <span m="3627470">V4,</span> <span m="3629260">basic,</span> <span m="3629545">we'll</span> <span m="3629830">color</span> <span m="3630140">that</span> <span m="3630620">in</span> <span m="3630760">d</span> <span m="3631180">plus</span> <span m="3631390">1</span> <span m="3631690">colors.</span> <span m="3633850">And</span> <span m="3633990">all</span> <span m="3634320">I</span> <span m="3634390">have</span> <span m="3634570">left</span> <span m="3635170">is</span> <span m="3635390">to</span> <span m="3635480">give</span> <span m="3635680">this</span> <span m="3635890">guy</span> <span m="3636090">color,</span> <span m="3636950">and</span> <span m="3637350">I'll</span> <span m="3637470">have</span> <span m="3637895">color</span> <span m="3638320">G.</span> <span m="3639260">Question?</span> <span m="3641360">No.</span> </p>
<p><span m="3643640">All</span> <span m="3643820">right.</span> <span m="3645600">So</span> <span m="3645820">by</span> <span m="3645990">induction</span> <span m="3646490">I've</span> <span m="3646620">colored</span> <span m="3647030">these</span> <span m="3647290">guys,</span> <span m="3647990">V1</span> <span m="3648320">to</span> <span m="3648420">V2,</span> <span m="3648820">and</span> <span m="3649020">d</span> <span m="3649230">plus</span> <span m="3649400">1</span> <span m="3649705">colors,</span> <span m="3650580">all</span> <span m="3650880">that</span> <span m="3651000">I</span> <span m="3651220">have</span> <span m="3651370">left</span> <span m="3651470">to</span> <span m="3651650">do</span> <span m="3651760">is</span> <span m="3652060">color</span> <span m="3652560">V</span> <span m="3652810">n</span> <span m="3653000">plus</span> <span m="3653500">1.</span> <span m="3656000">And</span> <span m="3656130">hopefully</span> <span m="3656720">we're</span> <span m="3656820">not</span> <span m="3657010">going</span> <span m="3657120">to</span> <span m="3657160">use</span> <span m="3657600">color</span> <span m="3658420">d</span> <span m="3658580">plus</span> <span m="3658850">2,</span> <span m="3659585">because</span> <span m="3660060">then</span> <span m="3660230">we</span> <span m="3660330">sort</span> <span m="3660560">of--</span> <span m="3660810">it</span> <span m="3660910">wouldn't</span> <span m="3661175">work.</span> <span m="3662450">We</span> <span m="3662460">got</span> <span m="3662650">to</span> <span m="3662700">use</span> <span m="3662920">one</span> <span m="3663070">of</span> <span m="3663130">the</span> <span m="3663200">first</span> <span m="3663480">d</span> <span m="3663971">plus</span> <span m="3664462">1.</span> </p>
<p></p>
<p><span m="3672330">All</span> <span m="3672450">right,</span> <span m="3672590">so</span> <span m="3672680">let's</span> <span m="3672970">look</span> <span m="3673170">at</span> <span m="3673250">V</span> <span m="3673610">n</span> <span m="3673860">plus</span> <span m="3674115">1.</span> <span m="3677110">And</span> <span m="3677320">let's</span> <span m="3677520">call</span> <span m="3677830">its</span> <span m="3677990">neighbors</span> <span m="3679160">in</span> <span m="3679340">G,</span> <span m="3680186">U1,</span> <span m="3682270">U2,</span> <span m="3685430">Ud.</span> <span m="3685780">It</span> <span m="3688490">has,</span> <span m="3688950">at</span> <span m="3689060">most</span> <span m="3689400">d</span> <span m="3689705">neighbors,</span> <span m="3690010">because</span> <span m="3690420">every</span> <span m="3690740">node</span> <span m="3690990">in</span> <span m="3691350">G</span> <span m="3691550">has,</span> <span m="3691630">at</span> <span m="3691920">most,</span> <span m="3692220">degree</span> <span m="3692430">d.</span> <span m="3694320">A</span> <span m="3694400">neighbor's</span> <span m="3694820">a</span> <span m="3694880">node</span> <span m="3695160">you're</span> <span m="3695300">adjacent</span> <span m="3695770">to.</span> </p>
<p><span m="3698840">All</span> <span m="3699205">right</span> <span m="3699570">so,</span> <span m="3701100">V</span> <span m="3701330">n</span> <span m="3701470">plus</span> <span m="3701790">1</span> <span m="3703120">has</span> <span m="3703690">at</span> <span m="3703850">most</span> <span m="3704160">d</span> <span m="3704470">neighbors,</span> <span m="3705310">is</span> <span m="3705800">adjacent</span> <span m="3705950">to,</span> <span m="3706430">at</span> <span m="3706620">most,</span> <span m="3706980">d</span> <span m="3707130">other</span> <span m="3707610">nodes.</span> <span m="3711220">Now</span> <span m="3712930">what</span> <span m="3713150">does</span> <span m="3713290">that</span> <span m="3713480">mean</span> <span m="3713690">about</span> <span m="3713900">the</span> <span m="3713990">color</span> <span m="3714410">I</span> <span m="3714470">can</span> <span m="3714630">use</span> <span m="3714990">on</span> <span m="3716170">V</span> <span m="3716350">n</span> <span m="3716460">plus</span> <span m="3716680">1?</span> <span m="3719540">What</span> <span m="3719680">do</span> <span m="3719790">I</span> <span m="3719880">know</span> <span m="3720120">about</span> <span m="3720350">what</span> <span m="3720520">color</span> <span m="3720840">I</span> <span m="3720890">can</span> <span m="3721660">use</span> <span m="3721990">for</span> <span m="3722060">that?</span> <span m="3724750">Yeah?</span> </p>
<p><span m="3725105">AUDIENCE: It</span> <span m="3725460">can't</span> <span m="3725725">be</span> <span m="3725990">any</span> <span m="3726260">of</span> <span m="3726730">the</span> <span m="3727200">colors</span> <span m="3727670">of</span> <span m="3728140">U1,</span> <span m="3728582">U2,</span> <span m="3729024">and</span> <span m="3729466">so</span> <span m="3729910">on.</span> </p>
<p><span m="3730070">PROFESSOR: It</span> <span m="3730410">can't</span> <span m="3730580">be</span> <span m="3730880">any</span> <span m="3731160">one</span> <span m="3731475">of</span> <span m="3732000">these</span> <span m="3732270">colors</span> <span m="3732560">that</span> <span m="3732850">were</span> <span m="3732960">assigned</span> <span m="3733410">here.</span> <span m="3734860">That's</span> <span m="3735100">true.</span> <span m="3736110">So</span> <span m="3736280">how</span> <span m="3736450">many</span> <span m="3736640">colors</span> <span m="3737110">got</span> <span m="3737310">ruled</span> <span m="3737660">out?</span> <span m="3739440">At</span> <span m="3739610">most</span> <span m="3739900">d,</span> <span m="3740812">and</span> <span m="3741270">how</span> <span m="3741420">many</span> <span m="3741660">am</span> <span m="3741790">I</span> <span m="3742150">working</span> <span m="3742520">with?</span> <span m="3743300">d</span> <span m="3743690">Plus</span> <span m="3744140">1.</span> <span m="3745560">So</span> <span m="3745740">I</span> <span m="3745810">got</span> <span m="3746030">one</span> <span m="3746210">left</span> <span m="3747300">that</span> <span m="3747440">I</span> <span m="3747510">can</span> <span m="3747670">use</span> <span m="3747890">safely.</span> <span m="3749580">OK.</span> </p>
<p><span m="3750920">So</span> <span m="3751080">this</span> <span m="3751330">means</span> <span m="3751830">there</span> <span m="3752040">exists</span> <span m="3753540">at</span> <span m="3753700">least</span> <span m="3753980">one</span> <span m="3754230">color</span> <span m="3755660">in</span> <span m="3755920">my</span> <span m="3756090">set</span> <span m="3756500">of</span> <span m="3756660">d</span> <span m="3756840">plus</span> <span m="3757150">1</span> <span m="3757600">colors.</span> <span m="3762560">It's</span> <span m="3762640">not</span> <span m="3763080">used</span> <span m="3766370">by</span> <span m="3766960">any</span> <span m="3767300">neighbor.</span> <span m="3773660">And</span> <span m="3773760">we're</span> <span m="3773880">going</span> <span m="3774000">to</span> <span m="3774050">give</span> <span m="3775190">V</span> <span m="3775390">n</span> <span m="3775520">plus</span> <span m="3775740">1</span> <span m="3775950">that</span> <span m="3776020">color.</span> </p>
<p></p>
<p><span m="3787330">All</span> <span m="3787460">right.</span> <span m="3787650">So</span> <span m="3787740">now</span> <span m="3788850">I've</span> <span m="3789040">colored</span> <span m="3789500">every</span> <span m="3789980">node</span> <span m="3790290">in</span> <span m="3790720">G,</span> <span m="3790850">the</span> <span m="3790950">n</span> <span m="3791170">plus</span> <span m="3791350">1</span> <span m="3791540">node</span> <span m="3791945">graph,</span> <span m="3792350">safely</span> <span m="3793490">using</span> <span m="3793850">a</span> <span m="3793920">total</span> <span m="3794310">of</span> <span m="3794420">d</span> <span m="3794620">plus</span> <span m="3794870">1</span> <span m="3795060">colors.</span> <span m="3798692">So</span> <span m="3799140">that</span> <span m="3799380">means</span> <span m="3799620">the</span> <span m="3799730">basic</span> <span m="3800190">algorithm</span> <span m="3804620">uses,</span> <span m="3805050">at</span> <span m="3805170">most,</span> <span m="3805800">d</span> <span m="3806010">plus</span> <span m="3806320">1</span> <span m="3806540">colors,</span> <span m="3809600">on</span> <span m="3809770">G.</span> <span m="3811496">That</span> <span m="3811920">means</span> <span m="3812160">P</span> <span m="3812460">n</span> <span m="3812880">plus</span> <span m="3813120">1</span> <span m="3813340">is</span> <span m="3813460">true--</span> <span m="3814630">whoops--</span> <span m="3816306">and</span> <span m="3816750">the</span> <span m="3817030">induction</span> <span m="3817170">is</span> <span m="3817340">complete.</span> <span m="3823050">Any</span> <span m="3823190">questions?</span> <span m="3824862">Yeah.</span> </p>
<p><span m="3825353">AUDIENCE: Could</span> <span m="3825844">you</span> <span m="3826335">also</span> <span m="3826826">start</span> <span m="3827317">from</span> <span m="3827808">the</span> <span m="3828299">other</span> <span m="3828790">way,</span> <span m="3829281">and</span> <span m="3829772">start</span> <span m="3830263">1,</span> <span m="3830754">go</span> <span m="3831245">to</span> <span m="3831736">2</span> <span m="3832227">nodes,</span> <span m="3832718">3</span> <span m="3833209">nodes</span> <span m="3833700">at</span> <span m="3834191">each</span> <span m="3834682">step</span> <span m="3835173">keeping</span> <span m="3835664">all</span> <span m="3836155">nodes</span> <span m="3836646">at</span> <span m="3837137">all</span> <span m="3837628">other</span> <span m="3838119">nodes.</span> <span m="3839101">[INAUDIBLE]</span> </p>
<p><span m="3842080">PROFESSOR: What</span> <span m="3842210">do</span> <span m="3842250">you</span> <span m="3842370">mean</span> <span m="3842550">by</span> <span m="3842660">keeping</span> <span m="3843070">all</span> <span m="3843450">nodes</span> <span m="3843870">connected?</span> </p>
<p><span m="3845308">AUDIENCE: [INAUDIBLE]</span> <span m="3853276">each</span> <span m="3853774">node</span> <span m="3854272">has</span> <span m="3854770">an</span> <span m="3855268">edge</span> <span m="3855766">connecting</span> <span m="3856264">to</span> <span m="3856762">each</span> <span m="3857260">other</span> <span m="3857650">one.</span> </p>
<p><span m="3858000">PROFESSOR: OK</span> <span m="3858350">so,</span> <span m="3858540">then</span> <span m="3858630">I</span> <span m="3858800">get</span> <span m="3858850">a</span> <span m="3859125">specific</span> <span m="3859400">graph.</span> <span m="3859720">I</span> <span m="3860040">start</span> <span m="3860290">with</span> <span m="3860410">this,</span> <span m="3861500">I</span> <span m="3861630">add</span> <span m="3861860">a</span> <span m="3861910">node</span> <span m="3862185">and</span> <span m="3862460">make</span> <span m="3862690">it</span> <span m="3863000">adjacent.</span> <span m="3864240">I</span> <span m="3864320">add</span> <span m="3864570">a</span> <span m="3864620">node</span> <span m="3864950">and</span> <span m="3865060">make</span> <span m="3865280">it</span> <span m="3865360">adjacent.</span> </p>
<p><span m="3865680">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3872410">PROFESSOR: Yeah.</span> <span m="3873904">So</span> <span m="3874391">you've</span> <span m="3875100">constructed</span> <span m="3875700">a</span> <span m="3875770">particular</span> <span m="3876200">graph.</span> <span m="3876620">This</span> <span m="3876830">is</span> <span m="3876940">actually</span> <span m="3877300">called,</span> <span m="3877900">for</span> <span m="3877980">the</span> <span m="3878110">n</span> <span m="3878440">nodes,</span> <span m="3878610">it's</span> <span m="3878960">called</span> <span m="3879306">Kn,</span> <span m="3880260">is</span> <span m="3880390">the</span> <span m="3880550">n</span> <span m="3880850">node</span> <span m="3881080">complete</span> <span m="3881410">graph,</span> <span m="3883720">also</span> <span m="3884050">called</span> <span m="3884330">a</span> <span m="3884400">clique,</span> <span m="3885010">like</span> <span m="3885160">a</span> <span m="3885230">clique</span> <span m="3885500">of</span> <span m="3885580">friends,</span> <span m="3887390">where</span> <span m="3887570">everybody</span> <span m="3887940">likes</span> <span m="3888230">everybody,</span> <span m="3889100">in</span> <span m="3889460">a</span> <span m="3889820">clique.</span> </p>
<p><span m="3892350">And</span> <span m="3892610">in</span> <span m="3892710">fact</span> <span m="3893440">for</span> <span m="3894010">n</span> <span m="3894310">here,</span> <span m="3894750">for</span> <span m="3894890">those</span> <span m="3895100">n</span> <span m="3895450">nodes,</span> <span m="3895660">what's</span> <span m="3895740">the</span> <span m="3895980">max</span> <span m="3896275">degree?</span> <span m="3898940">Max</span> <span m="3899250">degree</span> <span m="3899560">is</span> <span m="3899700">n</span> <span m="3899820">minus</span> <span m="3900140">1.</span> <span m="3901440">What's</span> <span m="3901860">the</span> <span m="3901950">chromatic</span> <span m="3902410">number</span> <span m="3902720">of</span> <span m="3902780">this</span> <span m="3902940">graph?</span> <span m="3906310">What's</span> <span m="3906550">the</span> <span m="3906640">minimum</span> <span m="3906980">number</span> <span m="3907180">of</span> <span m="3907230">colors?</span> </p>
<p><span m="3907485">[INTERPOSING VOICES]</span> </p>
<p><span m="3909210">PROFESSOR: And</span> <span m="3909490">they</span> <span m="3909610">all</span> <span m="3909780">have</span> <span m="3909980">to</span> <span m="3910060">be</span> <span m="3910150">different,</span> <span m="3911030">which</span> <span m="3911250">is</span> <span m="3911390">d</span> <span m="3911570">plus</span> <span m="3911860">1.</span> <span m="3912890">So</span> <span m="3913010">you</span> <span m="3913210">have</span> <span m="3913390">built</span> <span m="3913950">a</span> <span m="3914030">special</span> <span m="3914480">graph</span> <span m="3916420">for</span> <span m="3916600">which</span> <span m="3916910">the</span> <span m="3917220">optimum</span> <span m="3917400">of</span> <span m="3917590">number</span> <span m="3917840">colors</span> <span m="3918240">is</span> <span m="3918350">d</span> <span m="3918510">plus</span> <span m="3918760">1.</span> <span m="3919560">But</span> <span m="3919710">that</span> <span m="3919900">is</span> <span m="3920060">not</span> <span m="3920390">a</span> <span m="3920490">proof</span> <span m="3920800">that</span> <span m="3921110">this</span> <span m="3921410">is</span> <span m="3921500">true</span> <span m="3921740">for</span> <span m="3921920">all</span> <span m="3922080">graphs.</span> <span m="3923630">Because</span> <span m="3923790">you've</span> <span m="3924960">looked</span> <span m="3925160">at</span> <span m="3925260">a</span> <span m="3925320">particular</span> <span m="3925810">graph</span> <span m="3926293">here.</span> </p>
<p><span m="3926776">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="3930640">PROFESSOR: What's</span> <span m="3930830">that?</span> </p>
<p><span m="3931180">AUDIENCE: [INAUDIBLE]</span> <span m="3932400">It</span> <span m="3932700">means</span> <span m="3933190">that</span> <span m="3933680">you</span> <span m="3934020">can</span> <span m="3934190">still</span> <span m="3934350">use</span> <span m="3934570">your</span> <span m="3934760">less</span> <span m="3935050">than</span> <span m="3935210">or</span> <span m="3935310">equal</span> <span m="3935610">to</span> <span m="3935700">sign.</span> </p>
<p><span m="3936138">PROFESSOR: I</span> <span m="3936576">see,</span> <span m="3937014">so</span> <span m="3937452">you'd</span> <span m="3937890">add</span> <span m="3938180">a</span> <span m="3938470">node,</span> <span m="3938490">and</span> <span m="3938680">it's</span> <span m="3939120">only</span> <span m="3939270">connected</span> <span m="3939685">to</span> <span m="3940100">a</span> <span m="3940515">few</span> <span m="3940930">of</span> <span m="3941360">them.</span> </p>
<p><span m="3941470">AUDIENCE: No,</span> <span m="3941710">it's</span> <span m="3941770">connected</span> <span m="3941940">to</span> <span m="3942080">all</span> <span m="3942290">of</span> <span m="3942550">them,</span> <span m="3943015">but</span> <span m="3943480">it</span> <span m="3943660">still</span> <span m="3943760">implies</span> <span m="3944070">that</span> <span m="3944560">you</span> <span m="3944780">need</span> <span m="3945175">less</span> <span m="3945810">than</span> <span m="3945930">or</span> <span m="3946050">equal</span> <span m="3946360">to</span> <span m="3946640">the</span> <span m="3946970">colors.</span> <span m="3947050">It</span> <span m="3947280">turns</span> <span m="3947480">out</span> <span m="3947820">it</span> <span m="3947890">happens</span> <span m="3948040">to</span> <span m="3948330">be</span> <span m="3948690">equal</span> <span m="3949180">to.</span> </p>
<p><span m="3949250">PROFESSOR: Yes,</span> <span m="3949440">in</span> <span m="3949760">this</span> <span m="3949960">case</span> <span m="3950140">that's</span> <span m="3950486">right.</span> <span m="3950832">So</span> <span m="3951180">you've</span> <span m="3951370">made</span> <span m="3951490">an</span> <span m="3951910">argument</span> <span m="3951980">for</span> <span m="3952170">this</span> <span m="3952390">case</span> <span m="3952510">where</span> <span m="3952560">it</span> <span m="3952900">actually</span> <span m="3953230">is</span> <span m="3953560">equal,</span> <span m="3953740">but</span> <span m="3953890">that</span> <span m="3954060">only</span> <span m="3954350">worked</span> <span m="3954430">for</span> <span m="3954630">this</span> <span m="3954915">graph.</span> </p>
<p><span m="3955200">AUDIENCE: [INAUDIBLE]</span> <span m="3958392">worse</span> <span m="3958848">case.</span> </p>
<p><span m="3959310">PROFESSOR: It</span> <span m="3959480">is</span> <span m="3959700">the</span> <span m="3959800">worst</span> <span m="3960050">case,</span> <span m="3960680">so</span> <span m="3960870">it</span> <span m="3960980">meets</span> <span m="3962170">the</span> <span m="3962260">bound.</span> <span m="3962670">It</span> <span m="3962750">shows</span> <span m="3963030">you</span> <span m="3963140">cannot</span> <span m="3963540">improve</span> <span m="3963830">this</span> <span m="3963980">bound.</span> <span m="3964290">Yeah,</span> <span m="3964750">is</span> <span m="3964930">there</span> <span m="3965000">a</span> <span m="3965260">question</span> <span m="3965360">up</span> <span m="3965610">there?</span> </p>
<p><span m="3966052">AUDIENCE: All</span> <span m="3966494">I</span> <span m="3966936">was</span> <span m="3967378">going</span> <span m="3967820">to</span> <span m="3968130">say</span> <span m="3968440">is</span> <span m="3968900">that</span> <span m="3969060">you've</span> <span m="3969290">proved</span> <span m="3969710">it's</span> <span m="3969880">the</span> <span m="3970050">worst</span> <span m="3970450">case.</span> </p>
<p><span m="3970620">PROFESSOR: Right,</span> <span m="3970700">so</span> <span m="3970940">what</span> <span m="3971210">you've</span> <span m="3971400">done</span> <span m="3971650">here</span> <span m="3971930">is</span> <span m="3972210">you've</span> <span m="3972440">shown</span> <span m="3972670">that</span> <span m="3973080">I</span> <span m="3973130">could</span> <span m="3973320">not</span> <span m="3973760">make</span> <span m="3974110">that</span> <span m="3974610">theorem</span> <span m="3974740">any</span> <span m="3975180">stronger.</span> <span m="3975587">I</span> <span m="3975994">could</span> <span m="3976401">not</span> <span m="3976810">replace</span> <span m="3977140">it</span> <span m="3977540">with</span> <span m="3977923">d</span> <span m="3978306">here.</span> <span m="3978690">All</span> <span m="3978780">right.</span> <span m="3979030">Because</span> <span m="3979190">you've</span> <span m="3979300">given</span> <span m="3979460">an</span> <span m="3979530">example</span> <span m="3979960">where</span> <span m="3980080">I</span> <span m="3980140">can't</span> <span m="3980410">get</span> <span m="3980650">d</span> <span m="3980800">colors,</span> <span m="3981800">where</span> <span m="3981910">the</span> <span m="3981990">maximum</span> <span m="3982320">degree</span> <span m="3982530">is</span> <span m="3982700">d.</span> <span m="3983440">But</span> <span m="3983580">that</span> <span m="3983710">doesn't--</span> <span m="3984440">To</span> <span m="3984500">get</span> <span m="3984690">a</span> <span m="3984970">proof</span> <span m="3985250">for</span> <span m="3985370">a</span> <span m="3985500">theorem,</span> <span m="3985880">I</span> <span m="3985940">got</span> <span m="3986100">to</span> <span m="3986140">go</span> <span m="3986270">through</span> <span m="3986420">all</span> <span m="3986590">this.</span> <span m="3987110">That</span> <span m="3987360">wouldn't</span> <span m="3987610">give</span> <span m="3987730">me</span> <span m="3987860">a</span> <span m="3987920">proof</span> <span m="3988230">of</span> <span m="3988540">the</span> <span m="3988710">theorem.</span> </p>
<p><span m="3990480">They're</span> <span m="3991050">not</span> <span m="3991410">equivalent.</span> <span m="3992130">One's</span> <span m="3992540">an</span> <span m="3992630">upper</span> <span m="3992830">bound,</span> <span m="3993200">one's</span> <span m="3993630">an</span> <span m="3993710">existence</span> <span m="3994270">of</span> <span m="3994350">a</span> <span m="3994410">lower</span> <span m="3994680">bound.</span> <span m="3995660">This</span> <span m="3995870">shows</span> <span m="3996280">that</span> <span m="3996410">for</span> <span m="3996610">any</span> <span m="3996880">graph,</span> <span m="3999090">you</span> <span m="3999190">need</span> <span m="3999350">at</span> <span m="3999440">most</span> <span m="3999870">d</span> <span m="4000010">plus</span> <span m="4000280">1.</span> <span m="4000940">So</span> <span m="4001280">any</span> <span m="4001520">graph,</span> <span m="4002010">at</span> <span m="4002210">most.</span> <span m="4003990">That</span> <span m="4004310">shows</span> <span m="4005600">there</span> <span m="4005800">is</span> <span m="4006210">a</span> <span m="4006440">graph</span> <span m="4007525">that</span> <span m="4007850">you</span> <span m="4008070">need</span> <span m="4008360">at</span> <span m="4008520">least.</span> <span m="4010430">And</span> <span m="4010570">they</span> <span m="4010680">are</span> <span m="4010770">not</span> <span m="4010990">equivalent.</span> </p>
<p><span m="4012640">All</span> <span m="4012940">right.</span> <span m="4013230">One</span> <span m="4013600">is</span> <span m="4013890">for</span> <span m="4014120">all,</span> <span m="4014955">and</span> <span m="4015430">upper</span> <span m="4015650">bound.</span> <span m="4016610">The</span> <span m="4016710">other</span> <span m="4016940">is</span> <span m="4017170">there</span> <span m="4017410">exists</span> <span m="4018870">a</span> <span m="4018980">lower</span> <span m="4019250">bound.</span> <span m="4020830">So</span> <span m="4021010">different</span> <span m="4021890">in</span> <span m="4022020">two</span> <span m="4022210">ways</span> <span m="4022680">that</span> <span m="4022730">are</span> <span m="4022995">important.</span> </p>
<p><span m="4025590">This</span> <span m="4025970">kind</span> <span m="4026370">of</span> <span m="4026460">proof</span> <span m="4026920">is</span> <span m="4027140">very</span> <span m="4027580">typical</span> <span m="4028080">for</span> <span m="4028180">what</span> <span m="4028850">you'll</span> <span m="4028980">see</span> <span m="4029160">with</span> <span m="4029290">induction</span> <span m="4029710">in</span> <span m="4029790">graphs.</span> <span m="4031390">And</span> <span m="4031480">you'll</span> <span m="4031590">get</span> <span m="4031710">a</span> <span m="4032360">lot</span> <span m="4032550">of</span> <span m="4032630">practice</span> <span m="4033040">with</span> <span m="4033190">it.</span> <span m="4033340">Are</span> <span m="4033450">there</span> <span m="4033570">any</span> <span m="4033700">other</span> <span m="4034910">questions</span> <span m="4036030">on</span> <span m="4036890">this</span> <span m="4036910">proof?</span> <span m="4040200">OK.</span> </p>
<p><span m="4041870">All</span> <span m="4042370">right,</span> <span m="4042510">see</span> <span m="4042760">we've</span> <span m="4042980">seen</span> <span m="4043210">now,</span> <span m="4043400">by</span> <span m="4043540">that</span> <span m="4043730">example,</span> <span m="4044800">we</span> <span m="4045000">can't</span> <span m="4045320">improve</span> <span m="4045610">the</span> <span m="4045770">theorem.</span> <span m="4047330">In</span> <span m="4047490">some</span> <span m="4047860">cases,</span> <span m="4048520">though,</span> <span m="4049720">the</span> <span m="4049850">theorem</span> <span m="4050200">is</span> <span m="4050300">way</span> <span m="4050640">off,</span> <span m="4051120">for</span> <span m="4051270">some</span> <span m="4051570">graphs.</span> <span m="4052020">Can</span> <span m="4052170">anybody</span> <span m="4052530">think</span> <span m="4052800">of</span> <span m="4052880">a</span> <span m="4052940">graph</span> <span m="4053850">where</span> <span m="4054530">the</span> <span m="4054650">bound</span> <span m="4055100">we</span> <span m="4055210">get</span> <span m="4055420">from</span> <span m="4055590">the</span> <span m="4055940">theorem,</span> <span m="4056060">of</span> <span m="4056260">d</span> <span m="4056560">plus</span> <span m="4056810">1</span> <span m="4057125">colors,</span> <span m="4057850">is</span> <span m="4058030">way</span> <span m="4058400">off</span> <span m="4058750">from</span> <span m="4058920">the</span> <span m="4059050">actual</span> <span m="4059490">chromatic</span> <span m="4059910">number</span> <span m="4060160">you</span> <span m="4060280">need,</span> <span m="4060680">the</span> <span m="4060770">number</span> <span m="4060980">of</span> <span m="4061020">colors</span> <span m="4061370">you</span> <span m="4061580">need?</span> <span m="4062030">Yeah?</span> </p>
<p><span m="4062462">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4064190">PROFESSOR: What</span> <span m="4064340">is</span> <span m="4064440">it?</span> </p>
<p><span m="4064916">AUDIENCE: A</span> <span m="4065392">graph</span> <span m="4065868">[INAUDIBLE]</span> <span m="4069680">two</span> <span m="4070000">sets</span> <span m="4070320">of</span> <span m="4070801">[INAUDIBLE]</span> </p>
<p><span m="4075130">PROFESSOR: Good,</span> <span m="4075620">OK.</span> <span m="4076470">Yes,</span> <span m="4076810">so</span> <span m="4076920">what</span> <span m="4077200">if</span> <span m="4077290">we</span> <span m="4077510">did</span> <span m="4077730">this</span> <span m="4078120">graph.</span> <span m="4078220">Let</span> <span m="4078490">me</span> <span m="4078800">draw</span> <span m="4079110">it</span> <span m="4079576">out.</span> <span m="4083310">So</span> <span m="4083520">you've</span> <span m="4083680">got</span> <span m="4085546">a</span> <span m="4085968">bunch</span> <span m="4086390">of</span> <span m="4086740">nodes</span> <span m="4087080">here,</span> <span m="4087580">bunch</span> <span m="4087820">of</span> <span m="4087870">nodes</span> <span m="4088170">here.</span> <span m="4089600">And</span> <span m="4089850">every</span> <span m="4090230">node</span> <span m="4090540">here</span> <span m="4090820">is</span> <span m="4090910">connected</span> <span m="4091340">to</span> <span m="4091450">every</span> <span m="4091840">node</span> <span m="4092300">over</span> <span m="4092530">the</span> <span m="4092630">other</span> <span m="4092800">side.</span> <span m="4095600">And</span> <span m="4095800">if</span> <span m="4095870">this</span> <span m="4096029">is</span> <span m="4096149">an</span> <span m="4096250">n</span> <span m="4096420">no</span> <span m="4096609">graph,</span> <span m="4097069">and</span> <span m="4097170">I've</span> <span m="4097250">got</span> <span m="4097460">n</span> <span m="4097660">over</span> <span m="4097840">2</span> <span m="4098050">on</span> <span m="4098180">each</span> <span m="4098340">side,</span> <span m="4099680">what's</span> <span m="4100060">my</span> <span m="4100210">degree</span> <span m="4100620">here?</span> <span m="4106050">What's</span> <span m="4106149">my</span> <span m="4106260">max</span> <span m="4106560">degree</span> <span m="4106800">of</span> <span m="4106850">this</span> <span m="4107010">graph?</span> </p>
<p><span m="4107380">AUDIENCE: N</span> <span m="4107773">over</span> <span m="4108166">2.</span> </p>
<p><span m="4108560">PROFESSOR: N</span> <span m="4108770">over</span> <span m="4108990">2.</span> <span m="4110319">So</span> <span m="4110450">d</span> <span m="4110740">is</span> <span m="4110870">n</span> <span m="4111020">over</span> <span m="4111220">2.</span> <span m="4112680">What's</span> <span m="4113120">the</span> <span m="4113380">chromatic</span> <span m="4113790">number?</span> <span m="4114060">How</span> <span m="4114170">many</span> <span m="4114330">colors</span> <span m="4114630">do</span> <span m="4114740">I</span> <span m="4114819">need</span> <span m="4115020">for</span> <span m="4115100">this?</span> <span m="4116520">Two.</span> <span m="4118250">All</span> <span m="4118580">right,</span> <span m="4118790">so</span> <span m="4118890">d</span> <span m="4119109">plus</span> <span m="4119380">1</span> <span m="4119689">is</span> <span m="4119810">way</span> <span m="4120140">off</span> <span m="4120319">of</span> <span m="4120439">two.</span> <span m="4121380">There</span> <span m="4121569">is</span> <span m="4121670">a</span> <span m="4122010">even</span> <span m="4122279">worse</span> <span m="4122649">example.</span> <span m="4123613">Yeah?</span> </p>
<p><span m="4123966">AUDIENCE: That</span> <span m="4124319">graph</span> <span m="4124710">where</span> <span m="4125200">you</span> <span m="4125690">have</span> <span m="4126649">one</span> <span m="4126939">node</span> <span m="4127279">center</span> <span m="4127840">that's</span> <span m="4128130">connected</span> <span m="4128560">to</span> <span m="4128700">a</span> <span m="4128750">bunch</span> <span m="4129060">of</span> <span m="4129200">nodes</span> <span m="4129600">regularly</span> <span m="4130180">distributed</span> <span m="4130653">about.</span> </p>
<p><span m="4131126">PROFESSOR: Yeah,</span> <span m="4131600">the</span> <span m="4131720">star</span> <span m="4131974">graph.</span> <span m="4132564">All</span> <span m="4132899">right,</span> <span m="4133370">so</span> <span m="4133640">I</span> <span m="4133689">got</span> <span m="4133870">one</span> <span m="4134060">of</span> <span m="4134109">the</span> <span m="4134200">center,</span> <span m="4134620">I</span> <span m="4135060">got</span> <span m="4135399">n</span> <span m="4135649">minus</span> <span m="4135974">1</span> <span m="4136300">outside.</span> <span m="4138330">So</span> <span m="4138550">here</span> <span m="4138830">the</span> <span m="4138950">maximum</span> <span m="4139410">degree</span> <span m="4140620">is</span> <span m="4140850">n</span> <span m="4141020">minus</span> <span m="4141359">1,</span> <span m="4143050">just</span> <span m="4143330">like</span> <span m="4143450">a</span> <span m="4143520">complete</span> <span m="4143859">graph.</span> <span m="4145460">But</span> <span m="4146229">how</span> <span m="4146340">many</span> <span m="4146500">colors</span> <span m="4146800">do</span> <span m="4146890">I</span> <span m="4146960">need?</span> <span m="4148649">Two.</span> <span m="4150069">So</span> <span m="4150260">it's</span> <span m="4150359">even</span> <span m="4150580">worse</span> <span m="4151060">here.</span> </p>
<p><span m="4154319">All</span> <span m="4154479">right</span> <span m="4154710">now</span> <span m="4155700">what</span> <span m="4156050">about</span> <span m="4156740">the</span> <span m="4157979">basic</span> <span m="4158430">algorithm?</span> <span m="4158920">How</span> <span m="4159319">well</span> <span m="4159770">does</span> <span m="4159850">the</span> <span m="4159950">basic</span> <span m="4160430">algorithm</span> <span m="4160819">do</span> <span m="4161040">on</span> <span m="4161149">this</span> <span m="4161340">graph?</span> <span m="4164649">Or</span> <span m="4164840">to</span> <span m="4164880">the</span> <span m="4164970">vertices</span> <span m="4165500">some</span> <span m="4165810">way?</span> <span m="4167790">Color</span> <span m="4168074">on</span> <span m="4168359">one</span> <span m="4168660">[INAUDIBLE]</span> <span m="4168729">lowest</span> <span m="4169069">color.</span> <span m="4169630">How</span> <span m="4169940">many</span> <span m="4170460">colors</span> <span m="4170840">is</span> <span m="4170939">it</span> <span m="4171020">going</span> <span m="4171130">to</span> <span m="4171180">use?</span> </p>
<p><span m="4171620">AUDIENCE: Two.</span> </p>
<p><span m="4173840">PROFESSOR: Two.</span> <span m="4175240">It</span> <span m="4175390">doesn't</span> <span m="4175649">matter</span> <span m="4176689">the</span> <span m="4176770">vertices.</span> <span m="4178109">V1,</span> <span m="4180410">V2,</span> <span m="4181850">V3,</span> <span m="4182010">V4,</span> <span m="4183420">because</span> <span m="4183620">I'll</span> <span m="4183750">color</span> <span m="4184050">this</span> <span m="4184279">one</span> <span m="4184420">1.</span> <span m="4186619">What</span> <span m="4187049">am</span> <span m="4187479">I</span> <span m="4187649">going</span> <span m="4187800">to</span> <span m="4187859">call</span> <span m="4188069">that</span> <span m="4188270">one?</span> <span m="4189668">1.</span> <span m="4191950">Then</span> <span m="4192120">I</span> <span m="4192170">get</span> <span m="4192330">to</span> <span m="4192390">the</span> <span m="4192500">center,</span> <span m="4192840">what</span> <span m="4192960">am</span> <span m="4193029">I</span> <span m="4193069">going</span> <span m="4193160">to</span> <span m="4193220">color</span> <span m="4193550">it?</span> <span m="4194770">2.</span> <span m="4196360">And</span> <span m="4196480">now</span> <span m="4196670">all</span> <span m="4196940">the</span> <span m="4197080">arms,</span> <span m="4197480">what</span> <span m="4197620">do</span> <span m="4197650">they</span> <span m="4197760">get</span> <span m="4197920">colored?</span> <span m="4200130">They</span> <span m="4200270">all</span> <span m="4200560">get</span> <span m="4200710">1.</span> <span m="4201240">Whatever</span> <span m="4201570">order</span> <span m="4201820">you</span> <span m="4201960">pick,</span> <span m="4203610">you</span> <span m="4203700">get</span> <span m="4203880">two</span> <span m="4204000">colors.</span> </p>
<p><span m="4207270">All</span> <span m="4207320">right</span> <span m="4207630">so</span> <span m="4207730">now</span> <span m="4207990">there's</span> <span m="4208180">a</span> <span m="4208910">difference</span> <span m="4209270">between</span> <span m="4209580">the</span> <span m="4209620">theorem</span> <span m="4210160">just</span> <span m="4210340">gives</span> <span m="4210480">you</span> <span m="4210570">an</span> <span m="4210630">upper</span> <span m="4210850">bound,</span> <span m="4211100">it</span> <span m="4211180">says,</span> <span m="4211350">at</span> <span m="4211440">most,</span> <span m="4211930">d</span> <span m="4212070">plus</span> <span m="4212310">1</span> <span m="4212510">colors.</span> <span m="4213150">But</span> <span m="4213320">in</span> <span m="4213410">fact</span> <span m="4213710">the</span> <span m="4213880">algorithm</span> <span m="4214880">can</span> <span m="4215020">do</span> <span m="4215120">a</span> <span m="4215200">lot</span> <span m="4215500">better</span> <span m="4215700">than</span> <span m="4215870">that,</span> <span m="4216420">as</span> <span m="4216640">on</span> <span m="4216740">this</span> <span m="4216890">example.</span> <span m="4219050">So</span> <span m="4219090">the</span> <span m="4219230">algorithm</span> <span m="4219510">might</span> <span m="4219740">be</span> <span m="4219850">a</span> <span m="4219900">lot</span> <span m="4220130">better.</span> <span m="4222420">Everybody</span> <span m="4222740">see</span> <span m="4222960">that</span> <span m="4223490">what</span> <span m="4223630">we're</span> <span m="4223730">doing</span> <span m="4223960">here?</span> <span m="4224890">How</span> <span m="4225050">the</span> <span m="4225210">algorithm</span> <span m="4225590">is</span> <span m="4225700">better</span> <span m="4225950">than</span> <span m="4226100">the</span> <span m="4226180">bound</span> <span m="4226520">we</span> <span m="4226640">proved</span> <span m="4226900">by</span> <span m="4227000">the</span> <span m="4227120">theorem,</span> <span m="4227450">even</span> <span m="4227620">though</span> <span m="4227690">the</span> <span m="4227780">bound</span> <span m="4228160">was</span> <span m="4228340">pretty</span> <span m="4228680">good</span> <span m="4228840">for</span> <span m="4228960">some</span> <span m="4229170">graphs.</span> </p>
<p><span m="4231180">Now</span> <span m="4231380">it</span> <span m="4231490">turns</span> <span m="4231820">out--</span> <span m="4232715">I</span> <span m="4233180">mean</span> <span m="4233350">we're</span> <span m="4233440">not</span> <span m="4233620">going</span> <span m="4233710">to</span> <span m="4233760">win</span> <span m="4233910">a</span> <span m="4233950">million</span> <span m="4234220">dollars</span> <span m="4234640">for</span> <span m="4234730">this</span> <span m="4234910">algorithm.</span> <span m="4236110">And</span> <span m="4236260">in</span> <span m="4236340">fact,</span> <span m="4236680">this</span> <span m="4236760">algorithm</span> <span m="4237190">is</span> <span m="4237280">sometimes</span> <span m="4238120">very</span> <span m="4238600">bad.</span> <span m="4239880">And</span> <span m="4240400">a</span> <span m="4240500">really</span> <span m="4240830">bad</span> <span m="4241080">example</span> <span m="4241580">it's</span> <span m="4241750">very</span> <span m="4242170">close</span> <span m="4242550">to</span> <span m="4242640">this.</span> <span m="4244370">In</span> <span m="4244450">fact</span> <span m="4244680">actually</span> <span m="4244960">this</span> <span m="4245180">one,</span> <span m="4246010">let's</span> <span m="4246190">look</span> <span m="4246320">at</span> <span m="4247050">how</span> <span m="4247550">well</span> <span m="4247830">does</span> <span m="4248070">basic</span> <span m="4248470">do</span> <span m="4248610">one</span> <span m="4248760">this</span> <span m="4248950">one</span> <span m="4250870">here.</span> <span m="4254240">Make</span> <span m="4254480">some</span> <span m="4254680">ordering.</span> <span m="4256160">V1,</span> <span m="4257240">V2,</span> <span m="4259760">V3.</span> <span m="4262715">What's</span> <span m="4263140">the</span> <span m="4263360">basic</span> <span m="4263460">algorithm</span> <span m="4263830">going</span> <span m="4264270">to</span> <span m="4264340">do</span> <span m="4264520">on</span> <span m="4264630">this</span> <span m="4264810">complete--</span> <span m="4265520">it's</span> <span m="4265820">called</span> <span m="4266090">a</span> <span m="4266400">complete</span> <span m="4266850">bipartite</span> <span m="4267430">graph,</span> <span m="4267890">is</span> <span m="4268040">what's</span> <span m="4268180">this</span> <span m="4268450">called.</span> <span m="4268780">I'll</span> <span m="4269070">define</span> <span m="4269360">bipartite</span> <span m="4269430">in</span> <span m="4269470">a</span> <span m="4269840">minute--</span> <span m="4270300">but</span> <span m="4270360">what's</span> <span m="4270550">the</span> <span m="4270650">basic</span> <span m="4271030">algorithm</span> <span m="4271390">do</span> <span m="4271540">here?</span> <span m="4275690">Any</span> <span m="4275970">idea--</span> <span m="4276150">does</span> <span m="4276210">it</span> <span m="4276590">take</span> <span m="4276800">n</span> <span m="4276930">over</span> <span m="4277220">2</span> <span m="4277540">colors,</span> <span m="4278030">or</span> <span m="4278110">does</span> <span m="4278240">it</span> <span m="4278350">take</span> <span m="4278570">2?</span> <span m="4280860">Any</span> <span m="4280980">ideas?</span> <span m="4283170">2.</span> </p>
<p><span m="4284370">So</span> <span m="4284720">take</span> <span m="4284920">a</span> <span m="4284970">vertex,</span> <span m="4285460">and</span> <span m="4285660">the</span> <span m="4285730">first</span> <span m="4286050">one,</span> <span m="4286300">say</span> <span m="4286480">V1s</span> <span m="4286890">here,</span> <span m="4287170">get</span> <span m="4287360">C1.</span> <span m="4289650">As</span> <span m="4289810">long</span> <span m="4290000">as</span> <span m="4290110">I</span> <span m="4290170">keep</span> <span m="4290430">picking</span> <span m="4290720">vertices</span> <span m="4291210">over</span> <span m="4291420">on</span> <span m="4291520">this</span> <span m="4291740">side,</span> <span m="4293100">they're</span> <span m="4293250">going</span> <span m="4293350">to</span> <span m="4293390">get</span> <span m="4293570">C1.</span> <span m="4296430">As</span> <span m="4296580">soon</span> <span m="4296890">as</span> <span m="4297010">I</span> <span m="4297090">get</span> <span m="4297270">to</span> <span m="4297380">a</span> <span m="4297440">vertex</span> <span m="4297870">over</span> <span m="4298080">here,</span> <span m="4299560">what</span> <span m="4299750">color</span> <span m="4300020">does</span> <span m="4300220">it</span> <span m="4300380">have</span> <span m="4300470">to</span> <span m="4300725">get?</span> </p>
<p><span m="4301363">AUDIENCE: C2.</span> </p>
<p><span m="4302130">PROFESSOR: C2</span> <span m="4302760">because</span> <span m="4302920">it's</span> <span m="4303240">touching</span> <span m="4303620">the</span> <span m="4303700">very</span> <span m="4304020">first</span> <span m="4304270">one</span> <span m="4304390">we</span> <span m="4304480">had</span> <span m="4304650">here.</span> <span m="4306330">So</span> <span m="4306500">when</span> <span m="4306610">I</span> <span m="4306660">get</span> <span m="4306810">vertices</span> <span m="4307240">over</span> <span m="4307450">here,</span> <span m="4307670">they're</span> <span m="4307800">all</span> <span m="4307950">going</span> <span m="4308040">to</span> <span m="4308090">be</span> <span m="4308210">C2.</span> <span m="4308550">When</span> <span m="4308890">I</span> <span m="4308940">go</span> <span m="4309200">back</span> <span m="4309480">over</span> <span m="4309660">here,</span> <span m="4309860">they're</span> <span m="4310000">going</span> <span m="4310120">to</span> <span m="4310160">be</span> <span m="4310270">back</span> <span m="4310500">to</span> <span m="4310600">C1.</span> <span m="4312270">So</span> <span m="4312420">actually</span> <span m="4312810">basic</span> <span m="4313220">does</span> <span m="4313440">good</span> <span m="4313590">here</span> <span m="4313800">too,</span> <span m="4314770">gives</span> <span m="4314930">you</span> <span m="4315030">two</span> <span m="4315190">colors.</span> <span m="4316190">Yeah?</span> </p>
<p><span m="4316410">AUDIENCE:</span> <span m="4316909">[INAUDIBLE]</span> </p>
<p><span m="4326900">PROFESSOR: Ah,</span> <span m="4327230">those</span> <span m="4327680">two</span> <span m="4328100">aren't</span> <span m="4328550">connected.</span> <span m="4328620">But</span> <span m="4328910">this</span> <span m="4329300">case,</span> <span m="4330050">if</span> <span m="4330210">I've</span> <span m="4330380">got</span> <span m="4330530">a</span> <span m="4330560">vertex</span> <span m="4330990">over</span> <span m="4331190">here</span> <span m="4331480">it</span> <span m="4331860">is,</span> <span m="4331990">by</span> <span m="4332290">definition,</span> <span m="4332590">connected</span> <span m="4332980">to</span> <span m="4333040">the</span> <span m="4333130">vertex</span> <span m="4333590">over</span> <span m="4333810">here.</span> <span m="4334540">Because</span> <span m="4334970">every</span> <span m="4335400">possible</span> <span m="4335970">edge</span> <span m="4336280">is</span> <span m="4336610">here.</span> <span m="4337820">But</span> <span m="4337970">that's</span> <span m="4338210">a</span> <span m="4338260">great</span> <span m="4338550">idea.</span> <span m="4338870">What</span> <span m="4339120">if</span> <span m="4339270">they</span> <span m="4339380">weren't</span> <span m="4339920">all</span> <span m="4340900">connected,</span> <span m="4342510">that's</span> <span m="4342810">actually</span> <span m="4343060">a</span> <span m="4343090">great</span> <span m="4343390">idea.</span> </p>
<p></p>
<p><span m="4345590">In</span> <span m="4345860">fact,</span> <span m="4346210">the</span> <span m="4346320">nasty</span> <span m="4346840">example</span> <span m="4347340">for</span> <span m="4347430">the</span> <span m="4347530">basic</span> <span m="4347930">algorithm</span> <span m="4349270">is</span> <span m="4349450">very</span> <span m="4349680">much</span> <span m="4349890">like</span> <span m="4350120">that.</span> <span m="4350270">Let's</span> <span m="4350490">draw</span> <span m="4350968">it.</span> <span m="4355748">Because</span> <span m="4356710">so</span> <span m="4357010">far,</span> <span m="4357250">the</span> <span m="4357360">basic</span> <span m="4357790">algorithm</span> <span m="4358200">is</span> <span m="4358360">pretty</span> <span m="4358780">much</span> <span m="4359450">done</span> <span m="4359650">perfectly</span> <span m="4361430">on</span> <span m="4361570">all</span> <span m="4361700">the</span> <span m="4361780">graphs</span> <span m="4362080">we</span> <span m="4362170">looked</span> <span m="4362400">at</span> <span m="4364620">even</span> <span m="4364920">when</span> <span m="4365030">the</span> <span m="4365120">theorem</span> <span m="4366992">wasn't</span> <span m="4367460">tight.</span> <span m="4368870">So</span> <span m="4369040">here</span> <span m="4369370">is</span> <span m="4369480">a</span> <span m="4369540">nasty</span> <span m="4370100">graph.</span> <span m="4376140">And</span> <span m="4376310">it</span> <span m="4376790">is</span> <span m="4377020">very</span> <span m="4377450">close</span> <span m="4377790">to</span> <span m="4377870">the</span> <span m="4377930">graph</span> <span m="4378230">we</span> <span m="4378330">just</span> <span m="4378530">look</span> <span m="4378740">like,</span> <span m="4378930">where</span> <span m="4379150">all</span> <span m="4379550">the</span> <span m="4379740">edges</span> <span m="4380070">are</span> <span m="4380430">there.</span> </p>
<p><span m="4380910">In</span> <span m="4381040">this</span> <span m="4381250">case,</span> <span m="4381560">all</span> <span m="4381810">the</span> <span m="4382170">edges</span> <span m="4382320">are</span> <span m="4382670">there,</span> <span m="4383080">except</span> <span m="4383740">for</span> <span m="4383860">the</span> <span m="4383980">one</span> <span m="4384330">straight</span> <span m="4384650">across.</span> <span m="4387040">So</span> <span m="4387210">if</span> <span m="4387300">this</span> <span m="4387530">is--</span> <span m="4388810">the</span> <span m="4388990">edge</span> <span m="4389230">denotes</span> <span m="4389630">likes,</span> <span m="4390980">this</span> <span m="4391170">is</span> <span m="4391300">a</span> <span m="4391370">world</span> <span m="4391750">where</span> <span m="4391860">you</span> <span m="4392000">like</span> <span m="4392390">everybody</span> <span m="4393120">but</span> <span m="4393890">your</span> <span m="4394080">spouse.</span> <span m="4396950">All</span> <span m="4397170">right,</span> <span m="4397480">so</span> <span m="4397660">you</span> <span m="4398320">have</span> <span m="4398460">an</span> <span m="4398540">edge</span> <span m="4398740">to</span> <span m="4398900">every</span> <span m="4399180">one,</span> <span m="4399390">except</span> <span m="4399760">the</span> <span m="4399850">one</span> <span m="4400080">directly</span> <span m="4400490">across</span> <span m="4400910">from</span> <span m="4401060">you.</span> <span m="4405190">No</span> <span m="4405440">edge</span> <span m="4405660">there,</span> <span m="4406840">and</span> <span m="4407160">so</span> <span m="4407390">forth.</span> <span m="4409370">So</span> <span m="4409460">it</span> <span m="4409520">has</span> <span m="4409860">almost</span> <span m="4410470">every</span> <span m="4410770">edge,</span> <span m="4411010">but</span> <span m="4411110">it's</span> <span m="4411250">missing</span> <span m="4411910">these</span> <span m="4412170">edges.</span> </p>
<p><span m="4417620">Now</span> <span m="4418280">the</span> <span m="4418400">basic</span> <span m="4418880">algorithm</span> <span m="4419260">might</span> <span m="4419830">do</span> <span m="4419980">well</span> <span m="4420310">here.</span> <span m="4422440">What</span> <span m="4422800">would</span> <span m="4422910">be</span> <span m="4423050">a</span> <span m="4423110">good</span> <span m="4423410">ordering</span> <span m="4423930">for</span> <span m="4424030">this</span> <span m="4424240">graph</span> <span m="4426120">to</span> <span m="4426280">label</span> <span m="4426580">these</span> <span m="4426850">V1</span> <span m="4427350">through</span> <span m="4427500">Vn?</span> <span m="4428070">Yeah?</span> </p>
<p><span m="4428510">AUDIENCE: Go</span> <span m="4428984">through</span> <span m="4429458">everything</span> <span m="4429932">on</span> <span m="4430406">the</span> <span m="4430880">left</span> <span m="4431380">side,</span> <span m="4431630">and</span> <span m="4431830">then</span> <span m="4432328">the</span> <span m="4432826">right</span> <span m="4433324">side.</span> </p>
<p><span m="4434818">PROFESSOR: Yeah,</span> <span m="4435316">that's</span> <span m="4435814">right.</span> <span m="4443800">Because</span> <span m="4444270">then</span> <span m="4446110">color</span> <span m="4446440">1,</span> <span m="4448170">color</span> <span m="4448490">1,</span> <span m="4449650">color</span> <span m="4449960">1,</span> <span m="4450350">all</span> <span m="4450720">the</span> <span m="4450800">way</span> <span m="4450960">down.</span> <span m="4452480">One</span> <span m="4452760">color</span> <span m="4453030">for</span> <span m="4453120">the</span> <span m="4453190">left,</span> <span m="4455010">what</span> <span m="4455210">does</span> <span m="4455340">this</span> <span m="4455510">one</span> <span m="4455660">get?</span> <span m="4457460">Color</span> <span m="4457860">2,</span> <span m="4458330">because</span> <span m="4458480">it's</span> <span m="4458640">hooked</span> <span m="4458830">up</span> <span m="4458950">against.</span> <span m="4459540">And</span> <span m="4460290">these</span> <span m="4460690">all</span> <span m="4461070">get</span> <span m="4461270">color</span> <span m="4461570">2,</span> <span m="4462640">so</span> <span m="4462790">I've</span> <span m="4462930">used</span> <span m="4463250">two</span> <span m="4463400">colors.</span> <span m="4465670">Really</span> <span m="4465980">good.</span> <span m="4466260">Basic</span> <span m="4466650">algorithm's</span> <span m="4467020">looking</span> <span m="4467280">great.</span> </p>
<p><span m="4468630">Now</span> <span m="4468740">here's</span> <span m="4468980">a</span> <span m="4469070">harder</span> <span m="4469410">question.</span> <span m="4471470">Can</span> <span m="4471630">you</span> <span m="4471730">figure</span> <span m="4472080">out</span> <span m="4472300">a</span> <span m="4472430">bad</span> <span m="4473070">ordering</span> <span m="4474800">for</span> <span m="4474910">this</span> <span m="4475140">graph,</span> <span m="4475790">where</span> <span m="4476280">I</span> <span m="4476350">use</span> <span m="4476740">a</span> <span m="4476890">lot</span> <span m="4477320">more</span> <span m="4477530">than</span> <span m="4477700">two</span> <span m="4477900">colors.</span> </p>
<p><span m="4478385">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4482270">PROFESSOR: What</span> <span m="4482440">is</span> <span m="4482560">it?</span> </p>
<p><span m="4483040">AUDIENCE: It</span> <span m="4483455">starts</span> <span m="4483870">at</span> <span m="4484285">the</span> <span m="4484700">top</span> <span m="4485065">of</span> <span m="4485430">the</span> <span m="4485894">cross,</span> <span m="4486358">and</span> <span m="4486822">then</span> <span m="4487286">the</span> <span m="4487750">next</span> <span m="4488230">level</span> <span m="4488310">then</span> <span m="4488620">across.</span> </p>
<p><span m="4488930">PROFESSOR: Very</span> <span m="4489150">good.</span> <span m="4489560">V1,</span> <span m="4489750">V2.</span> <span m="4490740">Just</span> <span m="4491110">as</span> <span m="4491230">natural,</span> <span m="4491740">really,</span> <span m="4492035">if</span> <span m="4492330">think</span> <span m="4492540">about</span> <span m="4492850">it,</span> <span m="4493210">to</span> <span m="4493600">order</span> <span m="4493860">it</span> <span m="4493940">this</span> <span m="4494160">way.</span> </p>
<p></p>
<p><span m="4502670">All</span> <span m="4502910">right.</span> <span m="4503860">What</span> <span m="4504090">color</span> <span m="4504420">does</span> <span m="4504550">V1</span> <span m="4504990">get?</span> <span m="4506650">C1.</span> <span m="4507410">What</span> <span m="4507620">color</span> <span m="4507990">does</span> <span m="4508140">V2</span> <span m="4508350">get?</span> </p>
<p><span m="4508890">AUDIENCE: C1.</span> </p>
<p><span m="4509910">PROFESSOR: C1</span> <span m="4510160">because</span> <span m="4510510">it's</span> <span m="4510650">not</span> <span m="4510790">hooked</span> <span m="4511000">up</span> <span m="4511410">here.</span> <span m="4512250">What</span> <span m="4512620">color</span> <span m="4512850">does</span> <span m="4512990">V3</span> <span m="4513430">get?</span> </p>
<p><span m="4513735">AUDIENCE: C2.</span> </p>
<p><span m="4514505">PROFESSOR: C2.</span> <span m="4514970">What</span> <span m="4515830">about</span> <span m="4516020">V4?</span> </p>
<p><span m="4516480">AUDIENCE: C2.</span> </p>
<p><span m="4517920">PROFESSOR: C2.</span> <span m="4518470">It's</span> <span m="4518620">not</span> <span m="4518880">hooked</span> <span m="4519140">up.</span> <span m="4519670">It</span> <span m="4519760">can't</span> <span m="4520010">get</span> <span m="4520470">one,</span> <span m="4520950">because</span> <span m="4521080">that's</span> <span m="4521290">up</span> <span m="4521400">here.</span> <span m="4521560">And</span> <span m="4521820">it's</span> <span m="4521990">not</span> <span m="4522200">the</span> <span m="4522290">two,</span> <span m="4522460">so</span> <span m="4522580">it</span> <span m="4522630">gets</span> <span m="4522810">two</span> <span m="4523350">What</span> <span m="4523675">color</span> <span m="4524000">does</span> <span m="4524240">V5</span> <span m="4525020">get?</span> </p>
<p><span m="4525310">AUDIENCE: C3.</span> </p>
<p><span m="4526600">PROFESSOR: C3.</span> <span m="4527490">Because</span> <span m="4527660">it's</span> <span m="4527810">hooked</span> <span m="4528020">up</span> <span m="4528140">to</span> <span m="4528250">one</span> <span m="4528560">to</span> <span m="4528835">two.</span> <span m="4529450">V6 ?</span> </p>
<p><span m="4530806">AUDIENCE: C3.</span> </p>
<p><span m="4531610">PROFESSOR: C3,</span> <span m="4531970">it's</span> <span m="4532330">hooked</span> <span m="4532590">up</span> <span m="4532710">to</span> <span m="4532820">one</span> <span m="4533020">and</span> <span m="4533150">two,</span> <span m="4533370">but</span> <span m="4533560">not</span> <span m="4533930">three.</span> <span m="4534900">And</span> <span m="4535120">you</span> <span m="4535240">can</span> <span m="4535370">see</span> <span m="4535560">what's</span> <span m="4535760">happening</span> <span m="4536200">here.</span> <span m="4538560">All</span> <span m="4538840">the</span> <span m="4538920">way</span> <span m="4539080">down</span> <span m="4539390">here</span> <span m="4539680">he's</span> <span m="4539810">hooked</span> <span m="4540050">up</span> <span m="4540230">to</span> <span m="4540630">all</span> <span m="4540950">the</span> <span m="4541530">n</span> <span m="4541620">over</span> <span m="4541720">2</span> <span m="4541840">minus</span> <span m="4542120">1</span> <span m="4542280">colors.</span> <span m="4543165">So</span> <span m="4543510">he</span> <span m="4543680">also</span> <span m="4543945">takes</span> <span m="4544210">C</span> <span m="4544370">n</span> <span m="4544430">over</span> <span m="4544640">2.</span> <span m="4547110">So</span> <span m="4547470">if</span> <span m="4547530">you</span> <span m="4547630">pick</span> <span m="4547820">that</span> <span m="4548130">ordering,</span> <span m="4549190">not</span> <span m="4549570">so</span> <span m="4549760">good.</span> <span m="4550120">You</span> <span m="4550170">use</span> <span m="4550460">n</span> <span m="4550670">over</span> <span m="4550840">two</span> <span m="4551000">colors.</span> <span m="4552600">So</span> <span m="4552690">it</span> <span m="4552760">really</span> <span m="4553090">matters</span> <span m="4553810">the</span> <span m="4554000">ordering.</span> </p>
<p><span m="4555510">Now</span> <span m="4555620">I</span> <span m="4555700">should</span> <span m="4555980">say</span> <span m="4556250">graphs</span> <span m="4556760">like--</span> <span m="4557050">actually</span> <span m="4557400">any</span> <span m="4557560">questions</span> <span m="4558030">about</span> <span m="4558270">what</span> <span m="4558340">we</span> <span m="4558440">did</span> <span m="4558600">here?</span> <span m="4559700">About</span> <span m="4559940">this?</span> <span m="4562430">All</span> <span m="4562860">right,</span> <span m="4562940">now</span> <span m="4563050">I</span> <span m="4563100">should</span> <span m="4563340">say</span> <span m="4563570">that</span> <span m="4563690">graphs</span> <span m="4564190">like</span> <span m="4564400">this</span> <span m="4564560">have</span> <span m="4564710">a</span> <span m="4564770">special</span> <span m="4565160">name,</span> <span m="4566100">they're</span> <span m="4566230">called</span> <span m="4566680">bipartite</span> <span m="4567410">graphs.</span> <span m="4569470">And</span> <span m="4569560">that's</span> <span m="4569760">important</span> <span m="4570255">to</span> <span m="4570750">remember.</span> </p>
<p></p>
<p><span m="4584520">All</span> <span m="4584610">right,</span> <span m="4584800">so</span> <span m="4592480">a</span> <span m="4592540">graph</span> <span m="4593260">G</span> <span m="4597020">is</span> <span m="4597250">said</span> <span m="4597460">to</span> <span m="4597510">be</span> <span m="4597640">bipartite</span> <span m="4603780">if</span> <span m="4604080">the</span> <span m="4604180">vertices</span> <span m="4605640">can</span> <span m="4605890">be</span> <span m="4606140">split</span> <span m="4606750">into</span> <span m="4607010">two</span> <span m="4607210">sets,</span> <span m="4609370">or</span> <span m="4609510">partitioned,</span> <span m="4613316">and</span> <span m="4615780">we'll</span> <span m="4615960">call</span> <span m="4616280">them</span> <span m="4616610">a</span> <span m="4616720">left</span> <span m="4617040">set,</span> <span m="4617950">and</span> <span m="4618060">a</span> <span m="4618210">right</span> <span m="4618510">set,</span> <span m="4621360">so</span> <span m="4621620">that</span> <span m="4623660">all</span> <span m="4624070">the</span> <span m="4624230">edges</span> <span m="4631090">connect</span> <span m="4632920">a</span> <span m="4633050">node</span> <span m="4635260">in</span> <span m="4635420">the</span> <span m="4635530">left</span> <span m="4635910">set,</span> <span m="4637330">to</span> <span m="4637590">a</span> <span m="4637640">node</span> <span m="4640250">in</span> <span m="4640410">the</span> <span m="4640520">right</span> <span m="4640800">set.</span> <span m="4643880">So</span> <span m="4644120">in</span> <span m="4644180">fact,</span> <span m="4644550">a</span> <span m="4644650">lot</span> <span m="4644760">of</span> <span m="4644840">today</span> <span m="4645060">we've</span> <span m="4645190">been</span> <span m="4645300">looking</span> <span m="4645640">at</span> <span m="4645750">bipartite</span> <span m="4646400">graphs,</span> <span m="4646970">because</span> <span m="4647720">the</span> <span m="4647810">nodes</span> <span m="4648210">are</span> <span m="4648280">here.</span> <span m="4648500">Like</span> <span m="4648670">the</span> <span m="4648750">men,</span> <span m="4649590">and</span> <span m="4649820">the</span> <span m="4649900">women,</span> <span m="4650880">and</span> <span m="4651010">the</span> <span m="4651170">edges</span> <span m="4651440">only</span> <span m="4651670">go</span> <span m="4651970">from</span> <span m="4652120">the</span> <span m="4652200">left</span> <span m="4652660">to</span> <span m="4652860">the</span> <span m="4652960">right.</span> <span m="4653900">And</span> <span m="4654380">that</span> <span m="4654550">is</span> <span m="4654660">called</span> <span m="4654950">bipartite.</span> <span m="4655335">And</span> <span m="4655720">it's</span> <span m="4655870">called</span> <span m="4656110">bipartite</span> <span m="4656830">because</span> <span m="4657110">you</span> <span m="4657290">can</span> <span m="4657500">do</span> <span m="4657650">it</span> <span m="4657740">with</span> <span m="4657950">two</span> <span m="4658170">colors,</span> <span m="4659075">or</span> <span m="4659460">in</span> <span m="4659610">two</span> <span m="4659760">pieces.</span> </p>
<p><span m="4661820">So</span> <span m="4661860">you</span> <span m="4661970">don't</span> <span m="4662250">win</span> <span m="4662410">a</span> <span m="4662450">million</span> <span m="4662780">dollars</span> <span m="4663860">for</span> <span m="4664040">deciding</span> <span m="4664590">whether</span> <span m="4664840">or</span> <span m="4664890">not</span> <span m="4665110">a</span> <span m="4665160">graph</span> <span m="4665520">can</span> <span m="4665630">be</span> <span m="4665730">colored</span> <span m="4666050">in</span> <span m="4666340">two</span> <span m="4666635">colors.</span> <span m="4667600">That's</span> <span m="4667880">easy.</span> <span m="4668950">You'll</span> <span m="4669200">even</span> <span m="4669280">do</span> <span m="4669430">it</span> <span m="4669500">for</span> <span m="4669590">homework</span> <span m="4670050">one</span> <span m="4670300">of</span> <span m="4670410">these</span> <span m="4670790">times.</span> <span m="4671730">You</span> <span m="4671960">do</span> <span m="4672240">win</span> <span m="4672380">the</span> <span m="4672410">million</span> <span m="4672690">dollars</span> <span m="4673100">for</span> <span m="4673200">deciding</span> <span m="4673670">if</span> <span m="4673750">a</span> <span m="4673830">graph</span> <span m="4674200">can</span> <span m="4674330">be</span> <span m="4674430">colored</span> <span m="4674850">in</span> <span m="4675010">three</span> <span m="4675300">colors.</span> <span m="4676700">That's</span> <span m="4677040">really</span> <span m="4677300">hard</span> <span m="4678140">to</span> <span m="4678220">do.</span> </p>
<p><span m="4681090">Now</span> <span m="4681380">coloring</span> <span m="4681810">problems</span> <span m="4682270">come</span> <span m="4682400">up</span> <span m="4682610">in</span> <span m="4682760">all</span> <span m="4683000">sorts</span> <span m="4683790">of</span> <span m="4683930">applications.</span> <span m="4684800">You</span> <span m="4685243">know</span> <span m="4685686">with</span> <span m="4686240">this</span> <span m="4686380">company,</span> <span m="4686670">Akamai,</span> <span m="4687780">that</span> <span m="4688200">came</span> <span m="4688410">out</span> <span m="4688500">of</span> <span m="4688560">MIT,</span> <span m="4688950">we've</span> <span m="4689090">talked</span> <span m="4689380">about.</span> <span m="4690070">We</span> <span m="4690380">run</span> <span m="4690550">a</span> <span m="4690600">network</span> <span m="4691430">of</span> <span m="4691480">75,000</span> <span m="4692550">servers.</span> <span m="4694490">And</span> <span m="4694770">they're</span> <span m="4694950">used</span> <span m="4695230">to</span> <span m="4695290">distribute</span> <span m="4695710">content</span> <span m="4696150">on</span> <span m="4696240">the</span> <span m="4696370">internet,</span> <span m="4696860">and</span> <span m="4696970">so</span> <span m="4697150">forth.</span> <span m="4698280">And</span> <span m="4698660">we</span> <span m="4698870">have</span> <span m="4699060">to</span> <span m="4699170">deploy</span> <span m="4699540">a</span> <span m="4699610">new</span> <span m="4699840">version</span> <span m="4700210">of</span> <span m="4700280">our</span> <span m="4700410">software</span> <span m="4701160">on</span> <span m="4701230">those</span> <span m="4701420">servers,</span> <span m="4701800">pretty</span> <span m="4702030">much</span> <span m="4702260">every</span> <span m="4702550">week.</span> <span m="4703420">We're</span> <span m="4703600">pushing</span> <span m="4703980">new</span> <span m="4704120">software</span> <span m="4704660">out.</span> <span m="4705560">And</span> <span m="4706830">you</span> <span m="4706990">can't</span> <span m="4707340">deploy</span> <span m="4708570">on</span> <span m="4709210">every</span> <span m="4709460">server</span> <span m="4710100">at</span> <span m="4710210">the</span> <span m="4710300">same</span> <span m="4710630">time,</span> <span m="4711070">because</span> <span m="4711470">you've</span> <span m="4711590">got</span> <span m="4711760">to</span> <span m="4711850">take</span> <span m="4712180">down</span> <span m="4712550">a</span> <span m="4712610">server</span> <span m="4712950">to</span> <span m="4713050">deploy</span> <span m="4713330">new</span> <span m="4713480">software</span> <span m="4713900">on</span> <span m="4714030">it.</span> <span m="4714450">Got</span> <span m="4714580">to</span> <span m="4714630">take</span> <span m="4714800">it</span> <span m="4714860">out</span> <span m="4714970">of</span> <span m="4715010">commission.</span> </p>
<p><span m="4716050">And</span> <span m="4716210">so</span> <span m="4716300">we</span> <span m="4716410">can't</span> <span m="4716600">just</span> <span m="4716770">take</span> <span m="4716960">down</span> <span m="4717250">all</span> <span m="4717410">75,000</span> <span m="4718250">servers,</span> <span m="4718650">because</span> <span m="4718800">then</span> <span m="4718940">all</span> <span m="4719100">the</span> <span m="4719320">Facebook,</span> <span m="4719950">and</span> <span m="4720480">Netflix,</span> <span m="4720960">and</span> <span m="4721020">all</span> <span m="4721140">those</span> <span m="4721300">sites</span> <span m="4721750">would</span> <span m="4721900">stop.</span> <span m="4722410">That</span> <span m="4722580">would</span> <span m="4722680">be</span> <span m="4722790">bad.</span> <span m="4724840">And</span> <span m="4724980">we</span> <span m="4725090">can't</span> <span m="4725770">do</span> <span m="4725980">them</span> <span m="4726090">one</span> <span m="4726280">at</span> <span m="4726330">a</span> <span m="4726440">time,</span> <span m="4727340">because</span> <span m="4727640">there's</span> <span m="4727710">75,000.</span> <span m="4728060">And</span> <span m="4728760">it</span> <span m="4728910">takes</span> <span m="4729150">a</span> <span m="4729230">few</span> <span m="4729490">hours</span> <span m="4730180">for</span> <span m="4730360">each</span> <span m="4730510">one</span> <span m="4730780">to</span> <span m="4731310">get</span> <span m="4731450">the</span> <span m="4731530">traffic</span> <span m="4731930">off,</span> <span m="4732390">stop</span> <span m="4732800">it,</span> <span m="4733260">load</span> <span m="4733490">new</span> <span m="4733610">software,</span> <span m="4734070">and</span> <span m="4734150">turn</span> <span m="4734300">it</span> <span m="4734370">back</span> <span m="4734620">on.</span> <span m="4734980">And</span> <span m="4735420">it</span> <span m="4735590">would</span> <span m="4735710">take</span> <span m="4736090">us</span> <span m="4736500">years</span> <span m="4736910">to</span> <span m="4737030">do</span> <span m="4737180">one</span> <span m="4737410">software</span> <span m="4737880">install,</span> <span m="4738240">which</span> <span m="4738400">we</span> <span m="4738480">got</span> <span m="4738620">to</span> <span m="4738670">do</span> <span m="4738910">every</span> <span m="4739130">week.</span> </p>
<p><span m="4740610">So</span> <span m="4741140">we've</span> <span m="4741280">got</span> <span m="4741390">to</span> <span m="4741450">figure</span> <span m="4741710">out</span> <span m="4741820">a</span> <span m="4741870">schedule</span> <span m="4742360">for</span> <span m="4742560">how</span> <span m="4742740">many</span> <span m="4742930">servers</span> <span m="4743280">you</span> <span m="4743380">take</span> <span m="4743620">down</span> <span m="4743960">at</span> <span m="4744000">a</span> <span m="4744260">given</span> <span m="4744520">time,</span> <span m="4744610">and</span> <span m="4744790">which</span> <span m="4745290">ones.</span> <span m="4745530">And</span> <span m="4745680">it</span> <span m="4745780">turns</span> <span m="4746130">out</span> <span m="4746700">pairs</span> <span m="4747050">of</span> <span m="4747150">servers</span> <span m="4747590">have</span> <span m="4747780">certain</span> <span m="4748040">critical</span> <span m="4748370">functions.</span> <span m="4749690">So</span> <span m="4749730">there's</span> <span m="4749910">certain</span> <span m="4750240">pairs</span> <span m="4750680">of</span> <span m="4750770">servers</span> <span m="4751090">you</span> <span m="4751190">can't</span> <span m="4751520">take</span> <span m="4751700">down</span> <span m="4751930">at</span> <span m="4752000">the</span> <span m="4752070">same</span> <span m="4752330">time.</span> </p>
<p><span m="4754750">So</span> <span m="4754830">we</span> <span m="4754950">have</span> <span m="4755090">a</span> <span m="4755340">gigantic</span> <span m="4755970">75,000</span> <span m="4756900">node</span> <span m="4757130">coloring</span> <span m="4757520">problem,</span> <span m="4758590">where</span> <span m="4758670">there's</span> <span m="4758920">edges</span> <span m="4759330">between</span> <span m="4759680">servers.</span> <span m="4760035">Nodes</span> <span m="4760390">are</span> <span m="4760680">servers,</span> <span m="4760760">and</span> <span m="4761140">there's</span> <span m="4761340">an</span> <span m="4761420">edge</span> <span m="4761560">between</span> <span m="4761980">if</span> <span m="4762060">you</span> <span m="4762170">can't</span> <span m="4762920">install</span> <span m="4763250">new</span> <span m="4763380">software</span> <span m="4763850">at</span> <span m="4763920">the</span> <span m="4764000">same</span> <span m="4764260">time.</span> <span m="4765730">And</span> <span m="4765890">so</span> <span m="4766060">when</span> <span m="4766170">it</span> <span m="4766220">turns</span> <span m="4766560">out,</span> <span m="4766780">when</span> <span m="4766910">you</span> <span m="4767050">run</span> <span m="4767290">one</span> <span m="4767420">of</span> <span m="4767470">these</span> <span m="4767640">graph</span> <span m="4767940">coloring</span> <span m="4768270">algorithms</span> <span m="4769230">on</span> <span m="4769520">it,</span> <span m="4770410">you</span> <span m="4770540">could</span> <span m="4770720">do</span> <span m="4771030">it</span> <span m="4771170">with</span> <span m="4771420">eight</span> <span m="4771770">colors.</span> <span m="4772120">It</span> <span m="4772260">just</span> <span m="4772640">turns</span> <span m="4772910">out</span> <span m="4773080">that</span> <span m="4773250">way.</span> </p>
<p><span m="4774160">So</span> <span m="4774280">that</span> <span m="4774460">means</span> <span m="4774710">there's</span> <span m="4774990">eight</span> <span m="4775290">waves</span> <span m="4775980">of</span> <span m="4776080">install</span> <span m="4777070">that</span> <span m="4777450">go</span> <span m="4777560">on</span> <span m="4777770">to</span> <span m="4778070">the</span> <span m="4778150">network.</span> <span m="4778920">And</span> <span m="4779040">now</span> <span m="4779970">eight</span> <span m="4780260">times</span> <span m="4780680">a</span> <span m="4780740">few</span> <span m="4780930">hours</span> <span m="4781190">each</span> <span m="4781450">means</span> <span m="4781720">that</span> <span m="4781790">we</span> <span m="4781900">can</span> <span m="4782040">do</span> <span m="4782220">it</span> <span m="4782290">in</span> <span m="4782410">a</span> <span m="4782480">day,</span> <span m="4783450">and</span> <span m="4783550">you</span> <span m="4783640">can</span> <span m="4783770">manage</span> <span m="4784110">it.</span> </p>
<p><span m="4784672">You</span> <span m="4785114">know</span> <span m="4786000">on</span> <span m="4786110">a</span> <span m="4786150">much</span> <span m="4786350">smaller</span> <span m="4786800">scale,</span> <span m="4787780">the</span> <span m="4787870">same</span> <span m="4788130">problem</span> <span m="4788440">exists</span> <span m="4788940">for</span> <span m="4789310">register</span> <span m="4789760">allocation,</span> <span m="4790310">for</span> <span m="4790410">variables.</span> <span m="4792120">Here</span> <span m="4792340">you've</span> <span m="4792450">got</span> <span m="4792590">to</span> <span m="4792670">assign</span> <span m="4793060">every</span> <span m="4793250">variable</span> <span m="4793690">to</span> <span m="4793880">register.</span> <span m="4795520">But</span> <span m="4795660">you</span> <span m="4795730">can't</span> <span m="4796020">have</span> <span m="4796240">variables</span> <span m="4796770">that</span> <span m="4796860">are</span> <span m="4796960">active</span> <span m="4797350">at</span> <span m="4797400">the</span> <span m="4797490">same</span> <span m="4797810">time</span> <span m="4798840">associated</span> <span m="4799360">with</span> <span m="4799430">the</span> <span m="4799480">same</span> <span m="4799740">register.</span> <span m="4800900">And</span> <span m="4801040">you</span> <span m="4801140">want</span> <span m="4801310">to</span> <span m="4801360">minimize</span> <span m="4801940">the</span> <span m="4802020">number</span> <span m="4802250">of</span> <span m="4802320">registers</span> <span m="4802750">you</span> <span m="4802850">need.</span> </p>
<p><span m="4803850">So</span> <span m="4804020">again,</span> <span m="4804500">you</span> <span m="4804600">have</span> <span m="4804790">the</span> <span m="4804860">graph</span> <span m="4805180">coloring</span> <span m="4805500">problem.</span> <span m="4807110">The</span> <span m="4807200">number</span> <span m="4807470">of</span> <span m="4807530">colors</span> <span m="4808000">is</span> <span m="4808100">the</span> <span m="4808170">number</span> <span m="4808400">of</span> <span m="4808460">registers</span> <span m="4808930">you</span> <span m="4809260">need.</span> <span m="4810220">And</span> <span m="4810380">two</span> <span m="4810530">variables</span> <span m="4810845">can't</span> <span m="4811160">get</span> <span m="4811460">the</span> <span m="4811530">same</span> <span m="4811790">color</span> <span m="4812120">if</span> <span m="4812450">their</span> <span m="4812590">active</span> <span m="4812920">at</span> <span m="4813020">the</span> <span m="4813100">same</span> <span m="4813360">time,</span> <span m="4813640">so</span> <span m="4813780">you</span> <span m="4813930">put</span> <span m="4814020">an</span> <span m="4814160">edge</span> <span m="4814470">between</span> <span m="4814916">them.</span> </p>
<p><span m="4815810">The</span> <span m="4815890">most</span> <span m="4816150">famous</span> <span m="4816490">example</span> <span m="4816880">of</span> <span m="4816970">graph</span> <span m="4817300">coloring</span> <span m="4817720">is</span> <span m="4817890">the</span> <span m="4818130">map</span> <span m="4818540">coloring</span> <span m="4818870">problem,</span> <span m="4819160">with</span> <span m="4819250">the</span> <span m="4819320">four</span> <span m="4819590">coloring</span> <span m="4820520">theorem.</span> <span m="4821670">And</span> <span m="4821850">so</span> <span m="4821960">here,</span> <span m="4823650">every</span> <span m="4823920">country</span> <span m="4824350">is</span> <span m="4824470">a</span> <span m="4824530">node.</span> <span m="4825800">Adjacent</span> <span m="4826290">countries</span> <span m="4826690">have</span> <span m="4826850">an</span> <span m="4826930">edge</span> <span m="4827120">between</span> <span m="4827450">them,</span> <span m="4828010">because</span> <span m="4828170">you</span> <span m="4828260">don't</span> <span m="4828430">want</span> <span m="4828590">to</span> <span m="4828650">color</span> <span m="4829020">adjacent</span> <span m="4829460">countries</span> <span m="4829810">the</span> <span m="4829890">same</span> <span m="4830120">color,</span> <span m="4830520">or</span> <span m="4830630">you</span> <span m="4830980">can't</span> <span m="4831280">tell</span> <span m="4831480">they're</span> <span m="4831580">different</span> <span m="4831870">countries.</span> </p>
<p><span m="4834730">Now</span> <span m="4834890">the</span> <span m="4834990">last</span> <span m="4835320">example</span> <span m="4835645">we</span> <span m="4835970">can</span> <span m="4836210">talk</span> <span m="4836440">about</span> <span m="4836990">is</span> <span m="4838010">an</span> <span m="4838140">important</span> <span m="4838610">problem</span> <span m="4838910">in</span> <span m="4838970">communication</span> <span m="4839680">theory,</span> <span m="4840000">communication</span> <span m="4840610">networks,</span> <span m="4841690">where</span> <span m="4841870">again</span> <span m="4842250">coloring</span> <span m="4842680">comes</span> <span m="4842980">up.</span> <span m="4843630">Now</span> <span m="4843740">here</span> <span m="4844560">you</span> <span m="4844700">need</span> <span m="4844880">to</span> <span m="4844950">assign</span> <span m="4845340">frequencies</span> <span m="4846140">to</span> <span m="4846960">radio</span> <span m="4847390">stations,</span> <span m="4848100">or</span> <span m="4848290">the</span> <span m="4848380">cell</span> <span m="4848740">towers.</span> <span m="4849035">It</span> <span m="4849700">comes</span> <span m="4849950">up</span> <span m="4850070">in</span> <span m="4850170">mobile</span> <span m="4850490">networks,</span> <span m="4850980">or</span> <span m="4851100">just</span> <span m="4852120">in</span> <span m="4852310">with</span> <span m="4852450">radio</span> <span m="4852780">stations.</span> <span m="4854510">And</span> <span m="4855630">if</span> <span m="4855860">two</span> <span m="4856160">towers</span> <span m="4857320">have</span> <span m="4857510">an</span> <span m="4857620">overlapping</span> <span m="4858320">area,</span> <span m="4859370">they</span> <span m="4859490">can't</span> <span m="4859930">be</span> <span m="4860020">given</span> <span m="4860300">the</span> <span m="4860380">same</span> <span m="4860660">frequency,</span> <span m="4861290">so</span> <span m="4861410">you</span> <span m="4861540">get</span> <span m="4861740">collisions</span> <span m="4862490">between</span> <span m="4862820">the</span> <span m="4862910">towers.</span> <span m="4865540">And</span> <span m="4865720">frequencies</span> <span m="4866260">are</span> <span m="4866310">very</span> <span m="4866580">expensive.</span> <span m="4867110">Companies</span> <span m="4867530">pay</span> <span m="4867650">the</span> <span m="4867760">government</span> <span m="4868050">a</span> <span m="4868110">lot</span> <span m="4868360">of</span> <span m="4868420">money</span> <span m="4868750">to</span> <span m="4868840">get</span> <span m="4869070">certain</span> <span m="4869340">spectrum.</span> </p>
<p><span m="4871020">So</span> <span m="4871040">suppose</span> <span m="4871450">you</span> <span m="4871540">had</span> <span m="4871700">this</span> <span m="4871870">problem.</span> <span m="4873350">Here's</span> <span m="4873620">tower</span> <span m="4874010">A,</span> <span m="4874710">this</span> <span m="4874880">is</span> <span m="4875120">A's</span> <span m="4875520">range,</span> <span m="4876070">where</span> <span m="4876250">it</span> <span m="4876310">reaches.</span> <span m="4877205">Here's</span> <span m="4877810">tower</span> <span m="4878170">B,</span> <span m="4878850">so</span> <span m="4879490">it</span> <span m="4879820">overlaps</span> <span m="4880400">some</span> <span m="4880750">with</span> <span m="4881000">A.</span> <span m="4882630">Here's</span> <span m="4882950">tower</span> <span m="4883360">C.</span> <span m="4885180">Here's</span> <span m="4885660">tower</span> <span m="4887040">E.</span> <span m="4890270">And</span> <span m="4890490">here's</span> <span m="4890760">tower</span> <span m="4891600">D.</span> </p>
<p><span m="4896660">All</span> <span m="4896760">right</span> <span m="4896960">now</span> <span m="4897080">the</span> <span m="4897190">question</span> <span m="4897620">would</span> <span m="4897740">be,</span> <span m="4898280">how</span> <span m="4898520">many</span> <span m="4898740">radio</span> <span m="4899140">frequencies</span> <span m="4899740">do</span> <span m="4899820">you</span> <span m="4899910">need?</span> <span m="4901040">What's</span> <span m="4901290">the</span> <span m="4901370">minimum</span> <span m="4901740">number</span> <span m="4901990">of</span> <span m="4902060">frequencies</span> <span m="4902550">you</span> <span m="4902650">need</span> <span m="4902910">to</span> <span m="4903030">enable</span> <span m="4903950">all</span> <span m="4904160">the</span> <span m="4904270">towers</span> <span m="4904700">here?</span> <span m="4905916">We</span> <span m="4906300">could</span> <span m="4906390">make</span> <span m="4906490">that</span> <span m="4906660">be</span> <span m="4906840">a</span> <span m="4906990">graph.</span> <span m="4907480">There's</span> <span m="4907680">a</span> <span m="4907780">node</span> <span m="4908080">for</span> <span m="4908220">each</span> <span m="4908370">tower.</span> <span m="4909760">And</span> <span m="4909980">an</span> <span m="4910150">edge</span> <span m="4910470">between</span> <span m="4910910">towers,</span> <span m="4911410">if</span> <span m="4911520">they</span> <span m="4911680">overlap.</span> <span m="4914400">C</span> <span m="4914690">doesn't</span> <span m="4915040">overlap</span> <span m="4915480">with</span> <span m="4915610">B,</span> <span m="4917442">E</span> <span m="4917900">does.</span> <span m="4918650">E</span> <span m="4918910">overlaps</span> <span m="4919480">here.</span> <span m="4920970">And</span> <span m="4921190">then</span> <span m="4921540">D</span> <span m="4923030">overlaps</span> <span m="4923590">here.</span> </p>
<p><span m="4925060">So</span> <span m="4925190">how</span> <span m="4925310">many</span> <span m="4925500">frequencies</span> <span m="4925770">do</span> <span m="4926040">you</span> <span m="4926370">need</span> <span m="4928100">for</span> <span m="4928180">this</span> <span m="4928380">graph?</span> </p>
<p><span m="4929197">AUDIENCE: Four.</span> </p>
<p><span m="4930360">PROFESSOR: Four</span> <span m="4930930">would</span> <span m="4931090">work,</span> <span m="4932850">three</span> <span m="4933670">is</span> <span m="4933860">better.</span> <span m="4934580">Can</span> <span m="4935063">you</span> <span m="4935546">do</span> <span m="4936029">two?</span> <span m="4939900">No</span> <span m="4940280">you</span> <span m="4940370">can't</span> <span m="4940600">do</span> <span m="4940730">two,</span> <span m="4940930">because</span> <span m="4941090">you</span> <span m="4941190">got</span> <span m="4941400">here.</span> <span m="4942380">But</span> <span m="4942590">you</span> <span m="4942710">could</span> <span m="4942980">do</span> <span m="4943150">three.</span> <span m="4943820">You</span> <span m="4943920">could</span> <span m="4944060">do</span> <span m="4944220">one,</span> <span m="4945440">two,</span> <span m="4945930">three,</span> <span m="4947220">two,</span> <span m="4948190">one.</span> <span m="4949830">This</span> <span m="4950050">problem</span> <span m="4950370">comes</span> <span m="4950660">up--</span> </p>
<p><span m="4950945">AUDIENCE: [INAUDIBLE]</span> </p>
<p><span m="4951715">PROFESSOR: Did</span> <span m="4952200">I</span> <span m="4952400">screw</span> <span m="4952620">up?</span> <span m="4953010">Ooh,</span> <span m="4953310">no</span> <span m="4953420">I</span> <span m="4953560">can't</span> <span m="4953900">do</span> <span m="4954340">that.</span> <span m="4954660">One,</span> <span m="4955015">two,</span> <span m="4955370">yeah</span> <span m="4956120">much</span> <span m="4956330">better.</span> <span m="4956630">All</span> <span m="4957120">right,</span> <span m="4957610">this</span> <span m="4957990">problem</span> <span m="4958220">comes</span> <span m="4958420">up</span> <span m="4958560">all</span> <span m="4958750">over</span> <span m="4958870">the</span> <span m="4958960">place.</span> <span m="4959300">I'm</span> <span m="4959610">certain</span> <span m="4959840">you'll</span> <span m="4960000">see</span> <span m="4960460">it</span> <span m="4960790">sometime</span> <span m="4961130">in</span> <span m="4961180">your</span> <span m="4961300">career,</span> <span m="4961640">you'll</span> <span m="4961760">have</span> <span m="4961900">some</span> <span m="4962150">problem,</span> <span m="4962480">or</span> <span m="4962610">you're</span> <span m="4963020">scheduling</span> <span m="4963460">something,</span> <span m="4963710">and</span> <span m="4963830">it's</span> <span m="4963970">really</span> <span m="4964210">a</span> <span m="4964260">graph</span> <span m="4964560">problem</span> <span m="4964860">in</span> <span m="4964930">disguise.</span> <span m="4965960">OK</span> <span m="4966250">that's</span> <span m="4966500">it</span> <span m="4966600">for</span> <span m="4966700">today.</span> </p>
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